ilili:  li 


!  iiiiiiiliiili  hi 


III 


UCSB  LIBRARY 


THE 


MODERN   MECHANIC: 


SCIENTIFIC  GUIDE  AND  CALCULATOR 

COMPRISING 

RULES    AND    TABLES    IN    THE    VARIOUS 

DEPARTMENTS  OF   MECHANICAL 

SKILL  AND  LABOR 

BY   WILLIAM    GRIER, 

CIVIL   ENGINEER. 


How  have  we  obtained  this  great  superiority  over  th?se  poor  savages?  Becans* 
Science  has  been  at  work,  for  many  centuries,  to  diminish  th«  amount  of  our 
mental  labor,  by  teaching  us  the  easiest  mode  of  calculation. 

RESULTS  or  MACHINERY 


BOSTON: 
HIGGINS,  BRADLEY  AND  DAYTON, 

20  WASHINGTON   STREET. 


CONTENTS. 


Air  pump 222 

Air  vessel 228 

Animal  strength 273 

Arithmetic 17 

Artificers'  work 108 

Barometer 218 

Beam,  working  to  form 157 

Bramah's  press 180 

Catenary 99 

Collision 118 

Conic  sections 92 

Contraction,  marks  of 39 

Cotton  spinning 290 

Cube  root,  extraction  of 35 

Cycloid 98 

Drawing  instruments 80 

Drawing,  mechanical 86 

Eccentric 259 

Ellipse 94—97 

Floating  bodies 1 82 

Ply  wheel 260 

Forces,  central 146 

Forces,  parallelogram  of. 119 

Forcing  pump 227 

Fractions,  decimal 22 

Fractions,  vulgar 17 

Friction 275 

Geometry 57 

Governor .' 1-18 

( J ravity,  centre  of 134 

Gravity,  specific 182 

Gyration,  centre  of. 142 

Heat 238 

Heights,  measurement  of 219 

Horses  power 2o  t 

Hyd/odjh  arnica 195 

Hydrostatics ,   1 75 

Hyperbola 9(5—97 


Inclined  plane,  the 132 

Joists 163 

Journals 162 

Lever,  the 121 

Lifting  pump 226 

Machines 278 

Materials,  strength  of 148 

Materials,  weight  of 1 88 

Measures  and  weights 51 

Mechanics 115 

Mensuration 100 

Mill-weight's  table 212 

Motion,  accelerated 117 

Motion,  uniform 116 

Numbers,  compound 26 

Oscillation,  centre  of 137 

Parabola 94 — 97 

Parallel  motion 261 

Pendulum 137 

Percussion,  centre  of 141 

Pipes,  contents  of. 224 

Pneumatics 216 

Position 48 

Powers,  mechanical 121 

Powers  and  roots 33 

Progressions 44 

Proportion,  compound 43 

Proportion,  simple 41 

Pulley,  the 130 

Pumps 22 1 

Railways 266 

Rotation 142 

Screw,  the 133 

Sector,  the 83 

Shafts 159 

sliding  rule 36 

S  juare  ro'>t.  extr.u-tt'jii  of. ...     33 
Steam  .  .  243 


CONTENTS. 


1     P»g* 

Steam  engine 248 

Steam  vessels 269 

Suction  pump 223 

Syphon,  the 220 

Thermometer 238 

Timber,  measurement  of 107 

Water,  motion  of 195 

Water,  pressure  of 1 75 

Water  wheels v  204 


Wedge,  the 133 

Weights  and  measures 51 

Weight  of  materials 188 

Wheels 165 

Wheel  and  axle 124 

Wind 229 

Windmill,  horizontal 230 

Windmill,  vertical  . . '. 233 


TAB  L  ES. 


Alcohol,  vapour  of 247 

Capacities  for  heat 241 

Circular  segments 1 03 

Cohesion 150 

Crushing 151 

Drawing  paper 56 

Elasticity  and  strength  of  tim- 
ber   149 

Gauge  points 264 

Gravities,  specific 183 

Heat,  effects  of 240 

Iron  plate 189 

Iron  rod 191 

Iron,  Swedisjj,  weight  of. ....  188 

Iron,  wrougk-t,  weight  of. ....  1 88 

Lateral  strength 151 

Level,  difference  of. 209 

Machines,  power  of 289 

Metals,  weight  of. 189 — 190 

Mechanical  effect 288 


Millwright's 212 

Pipe,  cast  iron 191 

Pipes,  content  of. 226 

Pitch  of  wheels 168 

Platonic  bodies 1 05 

Polygons 101 

Proportions 47 

Shaft  journals 1 62 

Specific  heat 242 

Steam,  elasticity  of. 245 — 246 

Steam  vessels 270 

Traction,  force  of 269 

Water,  discharge  of. .  199—200— 
202—203 

Weight  of  materials 193 

Weights  and  measures 51 

Wheels 173 

I  Wheels,  teeth  of 175 

•  Wind,  force  of 230 

1  Windmill  sails 234 


INTRODUCTION. 


IT  is  our  intention,  in  these  introductory  panes,  to  make  a  tew 
observations  on  the  nature  of  scientific  knowledge,  which  may  be 
useful  to  the  younj  reader  in  enabling  him  to  understand 'more 
clearly  the  subjects  contained  in  the  volume,  and  in  guarding  him 
against  the  adoption  of  false  theory,  or  the  wasting  of  his  time 
in  inquiries  which  can  terminate  in  no  useful  result.  Such  intro- 
ductory observations  are  rendered  the  more  necessary,  as  a  correct 
knowledge  of  the  subjects  to  which  they  relate,  is  the  only  sure 
foundation  on  which  there  can  be  raised  a  solid  superstructure 
of  science. 

It  is  a  general  opinion  that  scientific  knowledge  is  entirely  dif- 
ferent from  all  other  kinds  of  knowledge  ;  or  that  it  requires  for 
its  cultivation  a  constitution  of  mind  only  to  be  met  with  here 
and  there  in  the  great  family  of  mankind  ;  and  what  is  said  of  the 
poet  is  also  thought  of  the  philosopher — th-.it  lie  is  burn,  not  made. 
All  men  are  certainly  not  equally  endowed  with  capacities  for  the 
acquisition  of  scientific  knowledge,  but  there  are  few  men  indeed 
who  are  totally  unprivileged.  The  man  who  would  relinquish 
scientific  pursuits  merely  because  he  had  no  hope  of  reaching 
the  eminence  of  a  Newton,  a  Watt,  or  a  Davy,  is  no  better  than 
him,  who,  in  despair  of  ever  obtaining  a  share  of  wealth  equal 
to  that  of  the  rich  inheritor  of  the  land,  would  cease  to  make  any 
honest  exertion  to  raise  himself  from  a  state  of  the  most  squalid 
wretchedness.  We  would  not  be  understood  by  this  to  bring  the 
acquisition  of  knowledge  into  invidious  comparison  with  the 
acquisition  of  wealth — the  one  is  in  every  case  a  godlike  employ- 
ment,  but  the  other  is  often  the  concomitant  of  vice. 

The  young  mechanic  should  be  made  well  aware  that  the 
knowledge  of  the  man  of  science  differs  from  the  knowledge  of 

1*  5 


6  INTRODUCTION. 

ordinary  men,  not  so  much  in  kind  as  in  degree  ;  and  the  know- 
ledge which  guides  the  little  boy  in  the  erection  of  his  summer- 
house,  constitutes  a  part  of  that  knowledge  which  guides  the  best 
architect  in  the  erection  of  the  most  splendid  edifice.  The  boy 
raises  his  paper  kite  in  the  air,  with  no  other  end  in  view  save 
his  own  amusement — he  has  learned  to  do  so  by  seeing  other 
boys  do  the  same,  and  by  trials  he  linds  that  the  kite  will  fly 
better  in  a  moderate  wind  than  in  a  perfect  culm,  and  that  the 
weight  at  the  tail  may  be  too  heavy  or  too  light,  and  he  regulates 
his  actions  accordingly  :  so  f;\r  he  is  a  little  philosopher.  /Y  man 
raises  a  kite  knowing  all  that  the  boy  know,  but  he  raises  it  with 
a  view  of  determining  the  state  of  the  atmosphere  so  far  as 
electricity  is  concerned,  for  which  purpose,  instead  of  employing 
the  hempen  cord,  which  was  sulFicient  for  the  purpose  of  the  boy, 
ne  employs  a  metallic  wire,  which  he  knows  by  experience  will 
conduct  the  electricity  from  the  clouds  to  the  earth,  and  thus 
effects  his  design.  In  this  respect  the  knowledge  of  the  man  is 
more  extensive  than  that  of  the  boy,  but,  this  additional  knowledge 
has  been  obtained  exactly  in  the  same  way  as  the  knowledge  of 
the  boy,  that  is  to  say,  by  experience.  Kven  the  Indian,  unlearned 
as  he  seems  to  be,  is  in  some  respects  a  philosopher.  He  sees 
daily  that  the  paddle  of  his  canoe  is  to  appearance  broken  when 
he  puts  it  into  the  water;  but  it  is  only  to  appearance,  for  by  re- 
peated trials,  he  finds  that  the  paddle  is  as  whole  when  in  the 
water  as  when  out  oi  it.  He  kno\vs  also,  by  repeated  trials, 
that  the  fish,  while  it  shoots  along  through  the  clear  flood,  does 
not  appear  to  be  where  it  really  is ;  for  though  the  most  unerring  of 
marksmen,  yet  if  he  throws  his  dart  directly  at  the  point  where 
the  fish  appears,  he  will  certainly  miss  it.  In  vain  will  he  try  to 
strike  the  fish  on  the  same  principles  as  he  strikes  the  bird  flying 
in  the  air;  but  he  finds,  that  when  he  directs  his  dart  to  a  line  which 
is  nearer  to  him  than  that  in  which  the  fish  seems  to  move,  he  will 
strike  the  fish.  The  Indian  remembers  the  circumstance  of  his 
paddle,  and  other  circumstances  of  a  like  kind,  and  concludes  that, 
when  bodies  are  viewed  through  water,  they  do  not  seem  to  be  in 
the  place  in  which  they  really  are.  When  he  knows  and  acts  upon 
this  principle,  he  is  a  man  of  science  so  far  as  this  is  concerned 
The  man  of  science,  indeed,  as  we  commonly  understand  that 
appellation,  knows  much  more  than  this :  he  knows  that  many 
other  substances  have  a  like  effect  in  changing  the  apparent 


INTRODUCTION.  7 

position  of  objects  wnen  seen  through  them  nat  one  r*  x>u>  «  a 
greater  and  another  a  less  change,  and  by  rer  rated  trmls  1e  as  ,er- 
rains  the  actual  amount  of  their  changes  by  measurement,  and 
can  subject  them  to  the  most  rigid  calculation ;  all  of  which 
knowledge  is  obtained  in  the  same  way  as  that  of  the  Indian,  but 
is  more  extensive. 

An  examination  of  facts  is  the  foundation  of  all  true  science  ; 
but  science  does  not  consist  in  a  mere  examination  of  facts. 
They  must  be  compared  with  each  other,  and  the  general  circum- 
stance of  their  agreement  carefully  marked.  When  we  have 
compared  several  facts  together,  and  find  that  there  is  one  general 
circumstance  in  which  they  agree,  this  one  circumstance  becomes, 
as  it  were,  a  chain  by  which  they  are  all  linked  together.  This 
general  circumstance  of  agreement,  when  expressed  in  language, 
is  what  is  called  a  law.  For  instance,  it  is  a  law  that  all  bodies, 
when  left  to  fall  freely,  will  tend  to  the  earth ;  and  this  law  has 
been  framed  by  us,  because  in  all  cases  which  we  have  examined 
this  has  been  the  case ;  and  the  term  gravity,  by  which  this  law 
is  designated,  is  nothing  else  than  a  name  invented  to  express  a  ( 
circumstance  in  which  we  have  found  innumerable  facts  to  agree. 
It  was  known  for  a  very  long  time  truflwater  would  not  rise  in  a 
sucking  pump  to  a  height  of  more  than  thirty-two  feet,  and  this 
was  said  to  take  place  because  nature  abhorred  a  vacuum.  The 
reason  given  was  afterwards  found  to  be  false,  yet  the  knowledge 
of  the  fact  was  exceedingly  useful  in  the  construction  of  pumps 
for  lifting  water.  About  the  middle  of  the  seventeenth  century, 
Toricelli,  the  pupil  of  Galileo,  made  experiments  on  the  subject, 
and  found  that  fluids  would  rise  in  tubes  or  in  sucking  pumps 
higher  in  proportion  as  they  were  lighter;  and  collecting  all  the 
facts  together,  he  concluded  that  the  fluids  were  forced  up  by  the 
pressure  of  the  atmosphere,  and  thus  laid  down  one  of  the  most 
hnportant  laws  of  physical  science.  A  collection  of  such  laws 
which  refers  to  some  particular  class  of  objects,  when  properly 
arranged,  becomes  what  is  called  a  theory.  Thus  we  see  that  a 
theory,  properly  so  called,  is  founded  on  an  examination  of  par- 
ticular facts,  and  of  course  cannot  refer  to  any  other  but  those 
facts  which  have  been  examined  ;  or,  if  it  is  attempted  so  to  do, 
it  is  no  longer  a  theory,  but  an  hypothesis  or  supposition. 
Hypotheses  although  they  ought  not  to  be  relied  upon,  are  never* 
theless  useful,  as  in  our  endeavours  to  discover  whether  they  t* 


8  INTRODUCTION. 

true  or  false,  we  may  at  last  ascertain  the  class  of  facts  to  which 
they  belong,  and  thus  arrive  at  the  true  theory. 

In  the  examination  of  facts,  it  is  to  be  observed,  that  we  must 
depend  on  the  information  derived  through  the  medium  of  the  five 
senses,  that  is,  the  senses  of  seeing — hearing — touching — tasting 
—and  smelling; — for  it  is  only  by  bodies  affecting  these  organ* 
fhat  the  properties  of  matter  become  known  to  us  ;  and  all  that 
the  mind  does  is  to  compare  and  classify  the  information  thus 
derived. 

It  is  a  common  error  to  suppose  that  many  of  our  greatest 
inventions  and  discoveries  were  made  by  accident.  Many 
wonderful  anecdotes  are  told  in  support  of  this  assertion ;  but 
the  very  circumstance  of  their  exciting  our  wonder  is  sufficient 
to  show  that  they  are  out  of  the  common  course  of  our  experience, 
and  that,  therefore,  before  they  are  received,  they  ought  to  undergo 
a  careful  examination.  A  multitude  of  facts  might  be  adduced 
to  prove  that  knowledge  is  more  regularly  progressive  than  is 
commonly  imagined.  Far  be  it  from  us  to  detract  from  the 
merits  of  those  great  men  who  have,  from  time  to  time,  benefited 
mankind  by  their  important  discoveries ;  but  from  a  survey  of 
the  history  of  science,  wfr*are  led  to  the  conviction,  that  where- 
ever  a  new  path  has  been  struck  out  in  the  great  field  of  truth, 
that  path  has  been  previously  prepared  by  former  inquirers.  Had 
Kepler  not  discovered  the  three  fundamental  laws  of  the  planetary 
motions,  it  is  highly  probable  that  the  Principia  of  Newton  never 
would  have  issued  from  the  pen  of  that  illustrious  man ;  and 
had  it  not  been  for  the  brilliant  discoveries  of  Dr.  Black  on  the 
subject  of  heat,  it  is  probable  that  Watt  never  would  have  made 
his  improvements  on  the  steam  engine,  that  invaluable  distributer 
of  power.  It  is  not  unlikely,  however,  from  the  state  of  know- 
ledge in  the  days  of  Newton,  that,  independent  of  the  exertions 
of  his  mighty  mind,  the  knowledge  contained  in  the  Principia 
would  soon  after  have  been  given  to  the  world  by  some  one  or 
more  individuals — and  the  like  may  be  said  of  the  inventions  of 
James  Watt. 

The  great  lesson  which  we  would  wish  the  young  mechanic  to 
.earn  from  these  observations  is — that  great  discoveries  are  nevei 
made  without  preparation — that  previous  knowledge  is  necessary 
to  turn  what  are  called  accidental  occurrences  to  good  account. 
And  when  he  is  told  that  the  law  of  gravitation  was  suggested 


INTRODUCTION.  9 

to  Newton  by  the  falling  of  an  apple  from  a  tree  in  his  garden 
or  that  the  invention  of  the  cotton  jenny  was  suggested  to  liar 
greave  hy  the  circumstance  of  a  common  spinning-wheel  conti- 
nuing in  its  ordinary  motion  while  in  a  state  of  falling  to  the 
ground — let  him  be  well  assured,  that,  had  the  minds  of  Newton 
and  Hargreave  not  been  previously  stored  with  knowledge,  these 
discoveries  never  would  have  been  made  by  them.  Apples  and 
spinning  wheels  bad  fallen  a  thousand  and  a  thousand  times,  but 
the  knowledge  necessary  to  turn  these  circumstances  -to  good 
account  was  first  concentrated  in  the  minds  of  these  two  illus- 
trious benefactors  of  mankind. 

In  Smith's  'Wen kh  uf  Nations  it  is  related  that  the  ingenious 
apparatus  for  opening  and  shutting  the  valves  of  the  steam  engine 
was  introduced  by  the  accident  of  an  idle  boy  having  fastened  a 
brick  as  a  counterweight  to  the  bandies  which  opened  and  shut 
tin.  valves,  and  thus  allowed  him  time  to  leave  the  machine  and 
go  to  play.  This  simple  trick  of  an  idle  boy,  it  is  said,  gave  rise 
to  the  apparatus  which  superseded  the  constant  attendance  of  a 
person  while  the  engine  was  at  work.  This,  however  romantic, 
is  not  the  fact — the  invention  originated  in  necessity,  no  doubt, 
but  it  was  begun  and  perfected  by  a  thorough  mechanic,  Mr.  H 
Brighton,  about  the  year  1717. 

While  we  are  on  this  subject  we  cannot  pass  over  another  very 
common  prejudice,  which  we  conceive  has  a  very  hurtful  tendency 
on  the  progress  of  the  young  mechanic.  We  allude  to  the  pride 
that  some  men  take  in  boasting  that  all  their  knowledge  is  ori- 
ginal ;  or  that  they  are  self-taught.  This  is,  in  other  words, 
stating,  that  no  assistance  has  been  taken  either  from  teachers  or 
books ;  and  goes  only  to  prove,  that  the  knowledge  of  the  indivi- 
dual so  circumstanced  must  be  very  limited  indeed.  The  unas- 
sisted exertions  of  one  man  must  be  very  feeble,  when  compared 
with  the  collected  exertions  of  the  many  who  have  gone  before 
him  in  the  career  of  discovery.  That  man  must  know  little  of 
geometry  who  has  not  availed  himself  of  the  use  of  Euclid's 
Elements,  or  some  work  of  a  similar  nature;  and  the  Elements 
of  Euclid  would  have  been  meager  and  confined,  had  he  not 
availed  himself  of  the  discoveries  of  his  contemporaries  and  pre- 
decessors. A  like  remark  may  be  made  on  the  cultivation  of 
every  department  of  knowledge  ;  and  to  those  whom  we  are  new 


10  INTRODUCTION. 

addressing  we  say — learn  from  others  all  that  you  possibly  can, 
and  when  you  have  done  so,  try  to  correct  and  improve  what  you 
have  obtained.  We  know  of  no  dishonourable  means  of  acquiring 
knowledge,  and  therefore  wherever  we  meet  it  we  are  disposed 
to  respect  it,  even  though  it  should  not  contain  one  particle  of 
originality,  if  such  be  possible ;  for  it  is  not  easy  to  conceive 
how  any  man  should  be  in  possession  of  useful  knowledge,  and 
not  make  some,  new  application  of  it ;  and  a  new  application  of 
an  old  principle  is  certainly  one  constituent  of  originality.  "With 
a  knowledge  of  what  others  have  done,  that  workman  will  bo 
less  likely  to  waste  his  time  in  enterprises  which  may  ruin  him 
by  their  failure,  or  in  speculations  which  are  unsupported  by  the 
principles  of  science. 

In  the  museum  of  the  mechanics'  class  of  the  university  founded 
hv  the  venerable  Anderson  of  Glasgow,  there  is  preserved  the 
model  of  a  machine  to  procure  a  perpetual  motion.  For  the  con- 
trivance and  execution  of  this  beautiful  specimen  of  workmanship, 
we  are,  we  believe,  indebted  to  an  ingenious  clock-maker  of  Dun- 
dee, who  has  proven  himself  a  master  in  the  use  of  his  tools. 
But  had  he  been  acquainted  with  the  first  principles  of  mechanics, 
or  with  the  nature  and  failure  of  the  various  attempts  which  had 
been  made  before  his  time  for  the  same  purpose,  he  would  have 
seen  the  utter  folly  of  his  enterprise,  and  would  have  spent  the 
seven  years  which  he  occupied  in  the  construction  of  this  truly 
beautiful  model  in  some  more  useful  employment.  These  seven 
years  might  have  been  devoted  to  the  construction  of  timepieces 
which  would  have  been  of  infinite  service  to  the  commerce  and 
navigation  of  his  country — in  guiding  the  lonely  mariner  when 
far  away  on  the  billow — in  determining  the  exact  distance  and 
direction  of  the  part  for  which  he  is  bound — whereas,  the  model 
of  his  perpetual  motion  is  preserved  in  the  museum  as  a  lasting 
monument  of  this  clock-maker's  ignorance,  perseverance,  and 
handicraft. 

It  is  another  common  error  to  suppose  that  genius  alone  can 
make  a  man  a  great  mechanic,  a  great  chemist,  or  a  great  any 
thing.  Some  one  makes  the  remark,  that  every  man  is  more 
than  half  humanity;  and  we  do  believe  that  the  differences  of 
the  degrees  of  knowledge  of  different  men  arise  more  from  their 
difference  of  application  than  from  original  differences  of  capa* 


INTRODUCTION.  11 

city.  Let,  therefore,  the  young  workman  earnestly  try  lo  learn 
and  we  do  assure  him  that  he  will  make  advances  which  will  he 
proportional  to  his  application. 

This  book  has  been  written  with  the  view  of  assisting  the 
young  workman  in  obtaining  a  knowledge  of  the  calculations 
connected  with  machinery.  The  first  part  is  devoted  to  such 
parts  of  arithmetic  as  workmen  generally  require,  and  in  which 
they  are  most  commonly  deficient.  Nor  is  this  deficiency  to  be 
wondered  at,  since  the  school  books  in  our  language  contain, 
generally  speaking,  no  explanation  of  the  nature  of  the  rules 
which  they  give,  and  are,  moreover,  embarrassed  with  so  many 
divisions  and  subdivisions,  that  the  mind  of  the  scholar  is  per- 
fectly perplexed,  nor  can  it  lay  hold  of  the  great  leading  principles 
which  pervade  the  whole  system.  As  this  is  the  great  instrument 
used  throughout  the  book,  we  have  endeavoured  to  make  its  use 
and  management  easily  understood.  The  examples  which  we 
have  given  are  indeed  few  and  simple ;  but,  if  carefully  consi- 
dered, they  will  be  found  sufficient  to  establish  the  principle. 
The  mere  habit  of  calculation  cannot  be  said  to  constitute  a 
knowledge  of  arithmetic  ;  it  is  easily  obtained,  but  is  of  no  avail 
without  the  principles.  This  is  well  illustrated  by  an  occurrence 
of  but  recent  date.  To  construct  a  set  of  mathematical  tables 
requires,  not  only  a  knowledge  of  principles,  but  also  immense 
calculation.  M.  De  Pronney  was  desired  by  the  government  of 
France  to  construct  a  very  large  set  of  such  tables ;  a  task  which 
would  require  the  labour  of  a  mathematician  for  many  years. 
But  Pronney  fell  upon  an  expedient  which  was  every  way  worthy 
of  a  man  of  science.  A  change  in  the  fashions  of  the  Parisians 
had  thrown  about  five  hundered  wig-makers  idle,  and  Pronney 
contrived  at  once  to  give  employment  to  these  barbers,  and  at 
the  same  time  to  serve  the  purposes  of  science.  He  digested  the 
principles  of  the  calculation  of  these  tables  into  short  and  simple 
rules,  and  printed  forms  of  them,  which  he  gave  into  the  hands 
of  these  workmen,  who,  in  a  few  months,  produced  a  set  of  tables, 
the  most  correct  and  extensive  that  ever  has  been  made.  The 
peruke-makers  may,  so  far  as  the  construction  of  the  tables  was 
concerned,  be  regarded  as  mere  machines,  under  the  guidance  of 
M.  de  Pronney.  The  same  principle  has  been  of  late  years  car- 
ried to  a  far  greater  extent  by  our  countryman,  Professor  Babbage, 
who  has  invented  a  machine  by  which  logarithms  and  astrono- 


12  INTRODUCTION. 

mica)  tables  may  be  calculated  and  printed  with  the  ttos  lunerring 
certainty,  thus  obviating  the  necessity  of  employing  either  calcu 
lators  or  compositors.  Let  not  these  statements  induce  you, 
however,  to  neglect  the  practice  of  calculation  ;  on  the  contrary, 
improve  yourself  in  it  wherever  you  can,  but  be  also  careful  to 
learn  the  principle. 

In  that  part  devoted  to  geometry,  we  have  given  such  informa- 
tion without  demonstration  as  was  necessary  to  the  right  under- 
standing of  the  rest  of  the  book  ;  and  the  like  may  be  said  of  the 
conic  sections,  mensuration,  and  useful  curves.  Thus  far  the 
book  may  be  said  to  be  a  compend  of  certain  branches  of  the 
mathematics.  It  is  hoped  that  the  reader,  to  whom  such  studies 
are  new,  will  not  be  contented  lo  stop  here ;  but  will  be  induced 
to  investigate  these  subjects  in  theory  ;  and  for  such  as  may  be 
desirous  of  entering  on  such  a  course  of  study,  where  there  is 
nothing  to  be  met  with  but  unsophisticated  truths  connected 
together  by  a  chain  of  the  most  beautiful  relations,  we  intend  to 
offer  a  few  words  of  well-meant  advice  as  to  the  order  and  means 
of  prosecuting  such  studies.* 

In  the  first  place,  let  the  Elements  of  Euclid  be  studied  so  far 
as  the  end  of  the  first  book,  in  the  course  of  which  it  should  be 
borne  in  mind,  that  there  is  nothing  really  difficult  to  be  met  with. 
The  greatest  difficulty  is,  we  believe,  this,  that,  to  a  proposition 
which  is  so  simple  as  to  be  almost  self-evident,  there  is  often 

*  In  a  very  creditable  work,  recently  published,  "  Stuart's  History  oi 
the  Steam  Engine,  "  it  is  stated  that  mathematics  is  not  necessary  to 
make  a  great  mechanic,  and  Watt  is  cited  as  an  instance.  The  instance 
chosen  is  most  unfortunate  for  the  author's  assertion.  Watt  was  de- 
scended from  a  family  of  mathematicians,  and  inherited  in  the  highest 
degree  the  genius  of  his  ancestors.  One  instance  will  sufficiently  prove 
this.  With  a  desire  to  determine  what  relation  the  boiling  point  bore  to 
the  pressure  of  the  atmosphere  on  the  surface  of  the  water,  he  made 
several  experiments  with  apothecaries'  phials,  and  having  found  the  rela- 
tion between  the  pressure  and  temperature  of  ebullition,  under  different 
circumstances,  he  laid  the  temperatures  down  as  abscissae,  and  the  pres- 
sures as  orrtinates,  and  thus  found  a  curve  whose  equation  gave  that  weF. 
known  formula,  the  equation  of  the  boiling  point.  No  man  but  a  mathe* 
matician  of  high  attainments  would  have  thought  of  such  a  method  of 
proceeding.  To  this  we  may  add,  that  mechanics  is  a  branch  of  mathe 
matics  ;  tor,  as  Sir  Isaac  Newton  has  defined  it,  "  mechanics  is  the  geo- 
metry of  motion." 


INTRODUCTION.  13 

attached  a  long  demonstration,  which  is  apt  to  lead  the  reader  to 
suppose  that  there  is  really  something  mysterious  in  it,  which  he 
does  not  understand.  This  proceeds  from  the  fact,  that  itoften 
requires  a  greater  deal  of  circumlocution  to  show  the  connection 
of  simple  propositions  with  first  principles,  compared  with  propo- 
sitions which  are  more  complex ;  but  we  have  no  hesitation  in 
saying,  that  if  the  steps  of  the  propositions  are  carefully  consi- 
dered, one  by  one,  they  will  be  easily  understood,  and  will  lead 
at  last  to  perfect  conviction ;  for,  as  Lord  Brougham  has  well 
observed,  "  Mathematical  language  is  not  only  the  simplest  and 
most  easily  understood  of  any,  but  the  shortest  also  ;"  and  Euclid 
has  transmitted  to  posterity  a  specimen  of  the  purest  mathemati- 
cal language.  Of  Euclid's  Elements,  there  are  various  editions. 
Those  of  Simpson  and  Playfair  are  generally  used  in  this  country, 
and  are  deservedly  popular.  That  of  Dr.  Thomson  is  a  very 
valuable  work,  and  very  correct.  But  we  beg  to  recommend  to 
the  workman  the  edition  of  Mr.  Robert  Wallace,  of  Glasgow, 
both  for  its  execution  and  cheapness.  The  demonstrations  are 
clear  and  short ;  many  new  propositions  are  added,  and  the  con- 
nection of  theory  with  practice  is  never  omitted  where  it  can  be 
introduced. 

When  the  first  book  of  Euclid  has  been  read,  the  study  of  al- 
gebra should  be  commenced,  on  which  subject  there  are  few  good 
treatises  to  be  found.  That  which  we  think  best  is  the  treatise 
of  Euler,  a  book  which  has  come  from  the  hand  of  a  master,  and 
is  therefore  characterized  by  great  simplicity.  Another  good 
book  is  the  treatise  of  Saunderson.  Let  either  of  these  works, 
or  others  if  they  cannot  be  had,  be  read  carefully  so  far  as  to 
equations  of  the  second  degree.  If  any  one  part  of  this  depart- 
ment can  be  said  to  be  difficult,  it  is  that  of  powers  and  roots, 
which  is  a  subject  of  the  greatest  importance;  and  should,  on 
that  account,  receive  the  most  careful  attention;  and,  if  the  trea- 
tise of  Euler  be  used,  we  have  no  hesitation  in  saying,  that  little 
difficulty  will  be  experienced.  Jt  may  be  necessary  to  observe, 
that  attention  should  be  paid  all  along  to  the  intimate  connection 
of  arithmetic  and  algebra,  which  will  tend  to  the  better  under- 
standing of  them  both.  Having  advanced  thus  far,  Euclid  must 
again  be  returned  to  ;  and,  after  revising  the  first  book,  re°.d  on 
to  the  sixth  inclusive.  Occasional  revision  of  the  algebra  is 
recommended,  and  an  advancement  as  far  as  equations  of  the 

2 


14  INTRODUCTION. 

third  degree  ;  after  which  Euclid  may  be  read  to  the  termination. 
The  study  of  trigonometry  may  then  be  introduced;  on  which 
subpect  we  have  various  works  of  various  merits.  The  treatise 
prefixed  to  Brown's  Logarithmic  Tables  may  be  employed ;  and 
when  it  is  understood,  and  the  management  of  the  logarithmic 
tables  acquired,  the  works  of  Gregory,  Lardner,  or  Thomson  may 
be  consulted ;  the  last  is  the  most  simple.  After  the  study  of 
trigonometry,  Simpson's  conic  sections  may  be  read  with  advan- 
tage. 

Perhaps  it  may  be  a  kind  of  relief  at  this  stage,  to  see  some- 
thing of  the  application  of  mathematics  to  mechanics,  and,  for 
this  purpose,  the  work  of  Keil  on  Physics,  or  the  article  Mecha- 
nics, Hutton's  Mathematics,  Tegg's  edition.  The  neat  little 
treatise  of  Mr.  Hay  of  Edinburgh  will  answer  the  same  purpose 
exceedingly  well.  But  for  the  purpose  of  obtaining  a  good  know- 
ledge of  theoretical  mechanics,  a  more  extensive  knowledge  of 
mathematics  than  we  have  hitherto  supposed  becomes  absolutely 
necessary.  A  knowledge  of  the  method  of  fluxions  and  fluents, 
or  the  differential  and  integral  calculus,  which  bear  a  strong  ana- 
logy to  each  other,  and  which  have  been  employed  for  similar 
purposes.  The  simplest  work  on  fluxions,  and  we  believe  the 
best,  is  the  treatise  of  Simpson ;  and  this  may  be  followed  by  a 
perusal  of  Thomson's  differential  and  integral  calculus.  With 
this  preparation  the  student  may  now  go  on  to  read  the  first  vo- 
lume of  Gregory's  Mechanics,  a  book  in  which,  we  believe,  he 
will  find  ample  satisfaction.  The  second  volume  of  this  excellent 
work  is  almost  entirely  popular,  and  can  cause  no  difficulty  what- 
ever. Another  work,  well  worthy  of  a  perusal,  is  that  of  Sir 
John  Leslie :  we  allude  to  his  Natural  Philosophy ;  a  book  which, 
though  neither  strictly  mathematical,  nor  strictly  popular,  yet 
contains  much  valuable  information  communicated  in  both  ways. 
Indeed  all  the  works  of  this  great  man,  although  much  has  been 
said  against  them  as  to  the  floridness  of  their  style,  will,  never- 
theless, be  found  to  amply  repay  the  trouble  of  a  perusal. 


THE 


MODERN    MECHANIC. 


ARITHMETIC. 


VULGAR  FRACTIONS. 

1.  IN  many  cases  of  division  after  the  quotient  is  ob- 
tained? there  is  a  remainder,  which  is  placed  at  the  end  of 
the  quotient,  above  a  small  line  with  the  divisor  under  it : 
thus — 88  divided  by  12  gives  the  quotient  7  and  remainder  4, 
which  is  written  12)  88  (7T42-     Now,  this  T\'s  caMe(l  a  frac- 
tion ;  and  it  is  written  in  this  way  to  show  that  4  ought  to 
be  divided  by  12  ;  and  in  all  cases  where  we  meet  with  num- 
bers written  in  this  form,  we  conclude  that  the  number  above 
the  line  is  to  be  divided  by  that  under  the  line.       This 
should  be  well  borne  in  mind,  as  it  is  of  the  greatest  use  in 
obtaining  a  clear  notion  of  fractions. 

2.  A  fraction  is  said  to  express  any  number  of  the  equal 
parts  into  which  one  whole  is  divided.  It  consists  of  two  num- 
bers— one  placed  above  and  the  other  below  a  small  line. 
The  upper  number  is   called  the   Numerator,  because   it 
numerates  how  many  parts  the  fraction  expresses ;  and  the 
under  number  is  called  the   Denominator,  because  it  ex- 
presses or  denominates  of  what  kind  these  parts  are ; — or, 
in  other  words,  the  denominator  shows  into  how  many  parts 
one  inch,  foot,  yard,  mile — one  whole  any  thing — is  sup- 
posed to  be  divided ;  and  the  numerator  shows  how  many 
of  these  parts  are  taken  :  as  ^  °f  a  foot.     The  denominator 
shows  that  the  foot   is  here  divided  into  12  equal   parts 
(inches  ;)  and  the  numerator  4,  shows  that  four  of  these 
parts  are  taken — (4  inches.) 

2*  17 


18  AKITHMETIC 

3.  If  the  numerator  had  been  equal  to  the  denominator, 
as  ^4,  then  the  value  of  the  fraction  would  have  been  one 
whole  (foot;)  and  the  numerator,    being  divided   by   the 
denominator,  gives  1  as  a  quotient.     In  the  fraction  }|  of  a 
foot,  the  numerator  is  greater  than  the  denominator,  and 
the  value  of  the  fraction  is  greater  than  one :  for  the  foot 
being  divided  into  twelve  equal  parts,  (inches,)  and  fourteen 
such  parts  (inches)  being  expressed  by  this  fraction,  its 
value   is  more   than  one  foot ;    and  the   numerator  being 
divided  by  the  denominator,  gives  \^ .     Again,  /^  of  a  foot 
is  just  six  inches,  or  one-half  foot ;  and  had  the  foot  been 
divided  into  two  equal  parts,  one  of  these  parts  would  have 
been  equal  to  T65,  or  5  is  equal  to  T*v.     From  this  we  may 
conclude,  that  when  the   numerator  is  equal  to,  less,  or 
greater  than  the  denominator,  the  value  of  the  fraction  is 
equal  to,  less,  or  greater  than  one  whole.     It  is,  then,  not  the 
numbers  which  express  the  numerator  and  denominator  of 
a  fraction,  but  the  relation  they  bear  to  each  other,  that 
determines  the  real  value  of  a  fraction.     £,  f ,  f ,  -,"2,  are  ah 
equal,  although  expressed  by  different  numbers, — the  deno- 
minators of  all  the  fractions  being  respectively  doubles  of 
their  numerators. 

4.  From  what  has  been  said,  it  will  easily  be  seen*  that, 
if  we  multiply  or  divide  both  terms  of  any  fraction  by  the 
same  number,  a  new  fraction  will  be  found,  equal  to  the 
first ;  thus,  £  ;  multiply  both  terms  by  2,  we  get  T8ff,  or 
divide  them  by  2,  f ,  and  these  again  by  2,  £.     All  who 
know  any  thing  of  a  common  foot-rule  will  understand  this, 
at  sight. 

5.  The  first  use  which  we  shall  make  of  the  principle  last 
stated,  is  to  bring  two  or  more  fractions  to  the  same  deno- 
minator, and  that  without  altering  their  real  values.     For 
example,  take  f  and  |  of  a  foot.     Multiply  both  terms  of  the 
first  fraction  f  by  the  denominator  of  the  second,  4  :  we  get 
•j%.     Next  multiply  both  terms  of  the  second  fraction  by  the 
denominator  of  the  first  fraction,  that  is,  |  by  3  :  the  result 
is  -j\.     Now  it  will^be  seen  (from  No.  4)  that  these  two 
fractions,  T8^  and  79^,  are  equal  to  the  two  f  and  |, — with 
this  additional  advantage,  however,  that  they  have  the  same 
denominator,  12  :  the  great  use  of  which  will  be  seen  here 
after.     A  like  process  is  employed  in  the  case  of  three  or 
more  fractions :  thus,  §,  |,  £, — multiply  the  terms  of  the 
first  fraction  by  4  and  5,  the  denominators  of  the  second 


VULGAK  FHACTIONS.  18 

and  third,  we  get  |-£  ;  next  multiply  the  second  |  by  3  and 
5,  the  denominators  of  the  first  and  third,  we  next  get  -££ ; 
lastly)  multiply  the  third  by  the  denominators  of  the  first  and 
second,  3  and  4,  we  get  £|.  It  will  be  useful  to  look  over 
what  we  have  done. — In  obtaining  the  numerators  of  the 
new  fractions,  we  have  multiplied  each  numerator  in  the 
former  fractions  by  all  the  denominators  except  its  own ; 
and  so  also  for  the  denominators.  But  3  multiplied  by  4, 
and  4  multiplied  by  3,  are  the  same  thing,  viz.  12  :  so,  like- 
wise, 3  multiplied  by  4  multiplied  by  5  is  60,  and  will  be  60  in 
whatever  order  we  take  them — 3  by  4  by  5,  or  4  by  3  by  5,  or 
5  by  3  by  4  ;  when,  therefore,  we  have  obtained  one  deno- 
minator, it  is  sufficient.  Hence  the  usual  rule  to  reduce 
fractions  to  a  common  denominator :  Multiply  each  nume- 
rator by  all  the  denominators  except  its  own  for  new  nume- 
rators, and  all  th»  denominators  together  for  the  common 
denominator. 

6.  We  are  now  prepared  to  add  two  or  more  fractions 
together.     It  is  very  easy  to  see  how  we  may  add  f  and  | 
of  an  inch,  and  that  their  sum  is  £ ;  but  it  is  not  quite  so 
evident  how  we  are  to  add  f  and  |  of  a  foot.     If  we  had 
them,  however,  of  one  denomination,  the  difficulty  would 
vanish.     By  No.  5,  bring  them  to  a  common  denominator — 
they  stand  thus :  T\  and  -^,  or  8  and  9  inches ;  add  the 
numerators,  and  under  their  sum  place  the  denominator,  }|- ; 
divide  the  numerator  by  the  denominator,  (No  1,)  the  quo- 
tient is  1TV,  or  one  foot  five  inches.     The  reason  of  bring- 
ing them  to  a  common  denominator  is,  that  we  cannot  add 
unlike  quantities  together :  and  we  do  not  add  the  denomi- 
nators, their  ojply  use  being  to  show  of  what  kind  the  quan- 
tities are.       The  rule,  then,  is — bring  the  fractions  to  a 
common  denominator,  add  the  numerators   together,  and 
under  their  sum  place  the  common  denominator. 

7.  In  subtraction  we  bring  the  fractions  to  a  common 
dei  ominator,  and  taking  the  lesser  from  the  greater  of  the 
two  numerators,  place  under  their  difference  the  common 
denominator.     The  reirson  of  this  may  be  easily  inferred 
from  (No.  6)  |  subtracted  from  5,  when  brought  to  a  com- 
mon denominator,  T6g  from  T8j  the1  difference  is  T2?,  equal  to 
i,  by  No.  4. 

8.  To  take  one  number  as  often  as  there  are  units  in 
another,  is  to  multiply  the  one  number  by  the  other.     To 
multiply  4  by  2,  is  to  take  the  number  four  two  times,  an 


20  ARITHMETIC. 

there  are  two  units  in  2  ;  and  to  multiply  4  by  |,  is  to  take 
four  one-half  times,  or  the  half  of  four,  as  there  is  only  half 
a  unit  in  the  fraction  5.  This  may  be  thought  so  simple, 
that  it  need  not  be  stated ;  but,  let  it  be  observed,  that  it 
explains  a  fact  in  the  multiplication  of  fractions,  which 
many  excellent  practical  arithmeticians  do  not  understand ; 
viz.  how  that,  when  we  multiply  by  a  fraction,  the  product 
is  less  than  the  number  multiplied.  If  the  fraction  5  is  to 
be  multiplied  by  ?,  (let  the  fractions  both  refer  to  an  inch,) 
this  is  taking  5  (inch)  ?  times,  or  taking  the  one-fourth  part 
of  one-half  inch,  which  is  one-eighth.  The  product  5  is 
obtained  by  this  simple  process  :  multiply  the  numerators 
together  for  a  new  numerator,  and  the  denominators  to- 
gether for  a  new  denominator  ;  the  new  fraction  will  be  the 
product.  That  this  is  true  in  general  may  be  shown  by 
taking  other  fractions,  thus:  £  of  f, — *he  product  by  the 
rule  is  ^,  which  may  be  simplified  by  dividing  the  nume- 
rator and  denominator  by  the  same  number,  on  the  principle 
of  No.  4 ;  if  4  be  the  divisor,  the  result  is  -£,  which  is  the 
same  as  •£•%.  Now,  that  %  is  the  real  product  of  |  by  f ,  may 
be  shown  thus  :  divide  a  line  AB 
into  six  equal  parts  ;  take  two  of  c  — 
these  parts,  and  join  them  by  A  — r 
CD.  Divide  CD  into  four  parts, 
and  it  will  be  seen  that  the  two  parts  of  this  line  CD  are  just 
equal  to  one  division  on  the  line  AB  ;  or  f  of  CD  is  equal 
to  £  of  AB  ;  so  that,,!  °f  f  ls  !•  ^'ne  rule>  then,  is  general. 

9.  Division  is  the  reverse  of  multiplication;  hence,  to 
divide  in  fractions, — invert  the  divisor,  and  proceed  as  in 
multiplication.     Thus,  to  divide  £  by  |,  insert  the  divisor 
4,  it  becomes  ^,  which,  multiplied  by  5  gives  5  multiplied  by 
},  equal  to  f ;  and  by  dividing,  to  make  the  fraction  less, 
we  obtain  f,  which,  by  No.  1,  is  just  2  or  twice.     This  is 
»he  quotient ;  and  it  is  easily  seen,  if  these  fractions  relate 
to  a  foot,  that  there  are  2  quarters  or  twice  £  of  a  foot,  in 
one-half  foot,  or  5. 

10.  We  have  now  endeavoured*  to  explain  the  nature  of 
the  fundamental  rules  of  vulgar  fractions,  as  simply  as  pos- 
sible ;  but  instances  often  occur,  where  it  is  necessary  to 
prepare  for  these  operations  ; — first,  where  whole  numbers 
are  concerned  ;  and  secondly,  where  the  fractions  are  large, 
and,  consequently,  not  so  easily  managed. 

11.  As  to  the  first,  where  whole  numbers  are  concerned. 


VULGAR    FRACTIONS.  21 

t  is  to  be  observed,  that  when  unit,  or  1,  is  used,  either  to 
multiply  or  divide  a  number,  it  does  not  change  the  value 
of  that  number.  Thus,  6  multiplied  by  1  is  6,  and  6  divided 
by  1  is  6.  According  to  the  principle  shown  in  No.  1,  we 
may  write  the  number  6  in  this  way,  f ,  without  altering 
its  real  value — with  this  advantage,  that  we  have  it  now  in 
the  form  of  a  fraction.  We  shall  illustrate  this  by  a  few 
examples,  and  show  that  numbers,  whether  whole  or  frac- 
tional, are  in  this  department  of  arithmetic  managed  by  the 
same  rules. 

Add  8  to  |,  here  we  write  them  £  and  I,  which,  brought 
to  a  common  denominator,  are,  3T2,  I — their  sum  is  3?s;  then 
by  No.  1,  divide  the  numerator  by  the  denominator,  we  get 
8|,  the  number  we  set  out  from.  7  j,  which  is  read  seven 
and  a  third,  may  on  the  same  principle  be  put  in  the  form 
of  a  common  fraction  :  for  it  is  7  wholes  added  to  5  part  of 
a  whole,  and  may  be  thus  written,  |  and  5,  equal  to  a-£  and 
whose  sum  is  \2;  divide  the  22  by  the  3,  the  result  is  7j, 
the  first  number.  This  very  simple  principle  is  often  used, 
and  is  embraced  in  the  following  rule — multiply  the  whole 
number  by  the  denominator  of  the  fraction,  add  the  nume- 
rator, and  under  the  sum  place  the  denominator. 

12.  When  the  fractions  are  very  large,  it  becomes  neces- 
sary to  bring  them  to  a  simple  form,  not  only  that  we  may 
more  easily  see  their  value,  but  that  they  may  be  more 
readily  operated  upon.  Thus,  j\  is  not  so  simple  nor  so 
easily  managed  as  TTj,  and  the  one  fraction  is  just  equal  in 
value  to  the  other;  for,  by  No.  4,  the  numerator  and  deno- 
minator of  762  being  both  divided  by  6,  gives  T'j.  Also, 
.ji^Pg.,  when  100  is  used  as  a  divisor,  gives  ?V-  Whenever 
we  can  find  a  number  which  will  divide  both  terms  of  the 
fraction  without  remainders,  we  ought  to  employ  it,  and  thus 
make  the  fraction  simpler  in  form,  though  of  exactly  the 
same  value.  The  divisor  thus  used  to  simplify  fractions,  is 
usually  called  the  common  measure,  and  may  frequently  :« 
found  at  sight,  although  sometimes  there  is  no  such  number 
at  all.  Thus,  in  ~ ,  it  is  seen  at  once  that  2  is  the  common 
measure  ;  but  in  the  fraction  |  no  such  common  measure 
can  be  found  :  consequently,  the  fraction  cannot  be  made 
more  simple.  Sometimes,  also,  two  or  more  numbers  will 
divide  the  fraction ;  thus,  f  may  be  divided  by  4  or  by  2— 
the  greatest  is  preferred,  because  it  brings  the  fraction  to 
the  lowest  terms  at  once.  When  this  cannot  be  obtained  at 


22  ARITHMETIC. 

sight,  the  following  rule  may  be  employed :  Divide  the 
greater  term  by  the  less ;  if  these  leave  any  remainder, 
divide  the  last  term  by  it ;  and  thus  go  on  dividing  the  last 
divisor  by  the  last  remainder,  and  that  divisor  which  leaves 
no  remainder  is  the  greatest  common  measure.  This  rule 
may  be  applied  to  the  following  example  : 

1 470  By  the  above  rule. 


2205       1470  )  2205  (  1 
1470 

~735)  1470  (2 
1470 

735  is  the  common  measure  ;  therefore, 

735  )  — —  (  f ,  the  simple  form  of  the  fraction 


DECIMAL  FRACTIONS. 

13.  LET  us  examine  the  number  3333,  (three  thousand, 
three  hundred,  thirty  and  three.)  The  same  figure  is  used, 
but  for  every  place  it  is  removed  towards  the  left,  its  value 
is  increased  ten  times  ;  and  consequently,  if  we  begin  at 
the  left  hand  side,  and  go  on  towards  the  right,  we  see  that 
every  figure  has  a  value  ten  times  less  than  the  same  figure 
placed  one  place  nearer  the  left, — each  number  expressing 
tenth  parts  of  the  number  next  it  to  the  left.  Hundreds  are 
just  tenth  parts  of  thousands ;  tens  are  tenth  parts  of  hun- 
dreds ;  and  units  are  tenth  parts  of  tens,  &c.  Now.  the 
same  3333,  with  a  point  placed  before  any  of  its  figures, 
would  still  have  the  same  property  of  each  figure  towards 
the  right,  having  a  tenth  part  of  the  value  it  would  have 
had  in  the  next  place  towards  the  left :  that  is  to  say,  the  point 
has  no  effect  in  altering  the  relative  value  of  the  figures  ;  but 
it  has  this  effect,  that  the  figure  which  stands  at  its  right 
hand  would  signify  units :  thus,  33-33,  where  we  have 
the  same  figures  as  before,  with  a  point  placed  betwixt  the 
middle  two,  and  from  what  has  been  said,  we  conclude  that 


DECIMAL    FRACTIONS.  23 

the  3  to  the  left  of  the  point  is  units.  From  this  it  follows 
that  the  next  3  on  the  right  of  the  point  is  tenth  parts  of 
unity,  and  the  3  following  that  again  tenth  parts  of  a  tenth 
part  of  unity,  or  hundredth  parts.  Had  it  been  written 
thus :  3.333,  the  last  three  to  the  right  of  the  point  would 
have  been  a  tenth  less  again,  &c ;  so  that  all  the  figures 
that  follow  the  point  to  the  right  are  less  than  units,  conse- 
quently, they  are  fractional ;  and  from  their  decreasing  by 
tenths  each  place,  they  are  called  Decimal  fractions — from 
the  Latin  word  decem,  ten.  Thus,  then,  T\  may  be  written  '3. 

14.  It  is  to  be  observed  here,  that  the  use  of  the  cipher 
(0)  is   in  decimals   quite   similar  to  what  it  is  in  whole 
r. ambers, — where  its  only  use  is  to  remove  some  figure  from 
the  units'  place,  and  therefore  alter  its  value  tenfold.     Thus, 
in  the  number  40,  the  cipher  of  itself  signifies  nothing,  but 
serves  to  remove  the  4  to  the  tens'  place.     Had  it  been  04— 
here  the  cipher  is  of  no  use,  because  there  is  no  figure  to 
remove  beyond  it  from  the  units'  place.    The  same  is  true  of 
any  number  of  units.     Now,  we  have  seen  that  '3  is  just  T3^ 
and,  from  what  has  been  said,  it  will  follow,  that  .03  is  three 
hundredth  parts,  or  T£7,  as  the  cipher  in  '03  removes  the  '3 
a  place  farther  from  the  units'  place  towards  the  right,  and 
(No.  13)  makes  it  ten  times  less  in  value  than  it  would  have 
been  had  it  been  one  place  nearer  the  left ;  or,  it  is  now 
tenth  parts  of  a  tenth  part.     For  the  same  reason  '003  is 
the  same  as  -j-^^. 

15.  The  number  33  is  read  thirty  and  three,  and  '33 
is  read  three  tenths  and  three  hundredths,  or  sometimes 
thirty-three  hundreds.     Now,  T3,  added  to  jfa  give  (No. 
6)  fVisV  which,  simplified,  is  T3^,  (No.  4.)    If  we  wished  to 
write  jjjVo  in  the  other  form,  it  is  done  simply  thus  :  point  0 
in  tenth's  place,  0  in  hundredth's  place,  and  3  in  thousandth's 
place  ;  that  is,  -003.     Take,  now,  -t\  and  T|^  ;  adding,  then, 
by  No.  6,  we  get-j^,0^,  simplified  ~£s,  which,  written  with  the 
point,  is  simply  *46.     We  may  now  see,  that  any  number 
placed  after  the  decimal  point  is  a  fraction ;  which  may  be 
expressed  by  a  numerator  which  is  that  number,  and  a  de- 
nominator consisting  of  1,  with  as  many  ciphers  annexed 
as  there  are  figures  in  the  numerator :  thus,  '3034  is  the 
same  thing  as  TVVoV 

16.  These  simple  statements  being  understood,  all  that 
follows  will  be  easy.     The  principle  being  kept  in  mind, 
that  the  numbers  to  the  one  side  of  the  point  have  the  same 


24  ARITHMETIC. 

relation  to  one  another  as  those  on  the  other, — e\ery  figure 
on  the  one  side  of  the  point  as  well  as  on  the  other,  being 
ten  times  greater  than  it  would  have  been  in  the  next  place 
to  the  right,  and  ten  times  less  than  in  that  to  the  left. 

17.  To  add  decimal  fractions,  we  proceed  just  as.in  whole 
numbers,  placing  units  under  units,  and  consequently  points 
under  points,  and  carrying  to  each  new  column  to  the  left, 
by  1  for  every  ten  in  the  column  already  added.     As  ^  may 
be  written  TS7  or  *5  ;  7£  may,  therefore,  be  written  7'5  ;  4^ 
may  be  written  4-5.    Now,  add  7*5  and  4-5  by 
the  rule  we  have  given,  and  we  will  obtain  a'         7"5 
result   which   must   be   correct, — as   may  be         4-5 
proved  by  principles  laid  down  in  the  former       12'0 
chapter.    Here  we  have  kept  the  points  under 
each  other,  and  put  a  point  in  the  answer  just  under  the 
others,  and  the  sum  is  12,  with  no  decimal  fraction.  Take  7  J 
and  bring  it  to  the  form  of  a  common  vulgar  fraction,  by  the 
principle,  No.  1 1 ,  and  it  will  be  y ;  do  so  likewise  with  4£  and 
we  get  | ;  they  have  a  common  denominator,  and  add  them 
by  No.  6,  we  have  2^4, — now,  this  fraction,  by  No.  4,  is  equal 
to  l^,  or  12,  the  same  as  before.    Take  now  135*7,  and  1-23, 
and  -764,  and  9-102,  and  8-003,  and  -035;   to  find  their 
sum.     Here  we  place,  as  before,  all  the  points  under  each 
other,  and  proceed  as  in  addition  of  whole  numbers,  carry- 
ing by  tens  and  pointing  the  sum  in  the  line  under  the  other 
points : 

135-7 

1-23 
•764 

9-102 

8-003 
•035 

154-834 


18.  Subtraction  is  managed  in  like  manner  as  in  common 
numbers,   the   same   attention    being   paid   to   the   points. 
Thus,   subtract  33-785  from   1967:32 ; 
they  are  placed  thus,  and  subtracted  as         1967*320 
in   whole   numbers,    the    point   in    the  33-785 

answer  being  placed  in  a  line  with  the         1933-535 
others.     It  is  to  be  observed,  that  there 
are  more  decimal  places  in  the  under  number  than  'n  th" 


DECIMAL    FRACTIONS.  25 

upper,  and  the  deficiency  may  be  supplied  by  adding  ciphers 
to  the  upper  line,  which,  as  there  is  no  significant  figure 
beyond,  does  not  alter  the  value  of  the  number. 

19.  Multiplication  of  decimal  fractions  is  performed  as  in 
whole  numbers,  paying  no  attention  to*the  points  until  the 
product  is  obtained,  when  we  point  off  as  many  places  from 
the  right  hand  side  of  the  product,  as  there  are  decimal 
places  in  both  the  multiplicand  and  the  number  which  mul- 
tiplies, or  multiplier.     Thus,  multiply 

36-42  by  4-7.     Here  -174  are  pointed  36-42 

off  as  decimals,  as  there  are  two  deci-  4*7 

mal  places  in  the  multiplicand  and  one  25494 

in  the  multiplier — in  all   three.     That          14568 
this   rule   is   correct,   may  be   inferred         17 1-174 
from  the  results  of  a  former  example  in 
No.  8.     Here  we  multiplied  4  by  £,  and  found  the  pro- 
duct to  be  2  :  now,  £  is  equal  to  T*T,  which  may  be  written 
•5 ;  then  let  us  multiply  4  by  -5,  as  directed 
above,  and  we  will  find  th«  same  result,  2 ; 
where,  by  principle  of  No.  14,   the   cipher 
being  pointed  off,  there  remains  2 — a  whole         2-0 
number. 

20.  Division  may  be  properly  defined,  the  finding  of  one 
number   (the   quotient),  such,   that  when   multiplied   by 
another  (the  divisor),  will  give  a  product  equal  to  a  third 
(the  dividend).     The  dividend  may  thus  be  viewed  as  the 
product  of  the  quotient  and  divisor ;  hence,  the  quotient  and 
divisor  should,  together,  contain  as  many  decimal  places  as 
the  dividend.     This  being  observed,  the  rule  will  be  easily 
followed :  Divide  as  in  whole  numbers,  and  when  the  quo- 
tient is  obtained,  point  off  from  the  right  as  many  places 
for  decimals  as  those  of  the  divisor  want  of  those  in  the 
dividend.     Divide  22-578  by  48-6, 

the  quotient  4-6f|.|  is  obtained  by  48-6)22'578(4-6f f.f 
common  division,  and  pointed  thus, 

because  the  divisor  wants  only  one  decimal  place  to  have 
as  many  as  the  dividend.  In  many  cases,  when  the  quo- 
tient is  obtained,  there  will  not  be  as  many  figures  as  make 
up  the  number  of  decimal  places  required  ;  here  we  must 
place  one  or  more  ciphers  betwixt  the  point  and  the  quo- 
tient figures,  so  as  to  make  up  the  number  required.  Thus, 
divide  1-0384  by  236,  the  quotient  is  44 — only  two  places, 
whereas  there  should  be  four  decimals  in  the  quotient ; 

8 


26  AUITHMETIC. 

because  there  are  four  in  the  dividend  and  none  in  tha 
divisor.  We,  therefore,  place  the  quotient  thus, — -0044 , 
and  to  prove  that  this  is  the  true  quotient,  we  have  only  to 
multiply  it  by  the  divisor,  and  the  product  being  the  same 
as  the  dividend,  the  operation  must  be  correct. 

21.  From  the  great  facility  with  which  decimal  fractions 
may  be  managed,  it  is  very  desirable  that  we  could  bring 
vulgar  fractions  to  the  same  form,  in  order  that  they  might 
more  easily  be  wrought  with.  Now,  this  may  be  done  on 
the  principles  already  laid  down  : — take  the  fraction  -§-,  and, 
on  the  principle  of  No.  4,  multiply  both  terms  by  1000,  it 
then  becomes  J°° °,  which  is  equal  to  I  ;  divide  (No.  4)  both 
numerator  and  denominator  by  8 ;  then  8)  £-°-$£  (TWo »  which 
last  fraction  is  expressed  in  the  decimal  notation  thus,  (on 
the  principle  of  No.  15,)  '125,  which,  from  the  way  it  has 
been  derived,  must  be  equal  to  G.  This  may,  however,  be 
found  more  immediately  thus  :  add  as  many  ciphers  to  the 
numerator  as  you  find  necessary,  and  divide  by  the  denomi- 
nator thus, — 8)1000(-125.  If  we  have  only  to  add  one 
cipher  before  we  get  a  quotient  figure,  we  put  a  point  in 
the  quotient ;  but  if  more,  then  we  put  as  many  ciphers  in 
the  quotient  after  the  point.  Thus,  -£j  ;  25)100('04,  and 
TJ  is  just  T^,  or  -04. 

22.  In  many  cases  the  quotient  would  go  on  without  end; 
but  it  is  to  be  observed,  that  it  is  not  necessary  to  continue 
any  operation  in  decimals,  at  least  in  mechanical  calcula- 
tions, beyond  three  or  four  places,  as  ten  thousandth  parts 
are  seldom  necessary  to  be  considered  in  practice.  For 
similar  reasons,  it  is  unnecessary  to  give  rules  for  repeating 
and  circulating  decimals  :  i.  e.  decimal  numbers,  when  the 
same  figures  recur  in  some  order — thus,  '3333,  or,  142142, 
&c.,  carry  them  to  four  places,  and  it  is  all  that  is  neces- 
sary. 

Other  applications  of  these  principles  will  be  found  in  the 
next  chapter,  on  compound  numbers. 


COMPOUND  NUMBERS. 

23.  IN  mechanical  calculations,  we  are  often  concerned 
with  weights  and  measures,  and  it  is  necessary  to  know  how 
to  operate  with  the  numbers  which  express  these.  The  rule 


COMPOUND    NUMBKRS.  27 

given  in  books  of  arithmetic  are  generally  very  long,  and, 
therefore,  not  very  easily  understood  ;  yet  the  steps  of  the 
operation  are  simple.  We  shall  therefore  show  the  mode  of 
procedure,  in  some  very  easy  examples,  and  the  reader  will 
find  no  difficulty  in  applying  the  principles  he  may  thus  im- 
bibe to  cases  more  complex.- 

24.  If  we  have  to  add  9  yards  2  feet  6    J'ds-      feet      «"*• 
inches,  to  2  yards  1  foot  3  inches,  8  yards 

0  feet  1 1  inches,  long  measure.   Then  we       ^ 
must  in  this,  as  in  all  other  cases  of  com- 
pound addition,  arrange  them  in  order,    -20         1          8 
the  greater  towards  the  left  hand,  and  the  lesser  towards 
the  right ;  and  there  must  be  a  column  for  every  denomina- 
tion of  weight  or  measure,  in  which  column  the  respective 
quantities  must  stand,  so  that  feet  will  stand  under  feet,  inch- 
es under  inches,  pounds  under  pounds,  and  ounces  under 
ounces,  &c.     Add  now  the  column  toward  the  right,  which 
in  this  example  amounts  to  20  inches,  or  1  foot  8  inches,  we 
therefore  put  down  the  8  inches  under  the  column  of  inches, 
and  add  the  1  foot  to  the  column  of  feet,  which  comes  to  4 
feet ;  that  is,  1  yard  and  1  foot.    The  1  foot  is  put  down  un- 
der the  column  of  feet,  and  the  1  yard  is  added,  or  carried,  as 
it  is  usually  called,  to  the  column  of  yards,  whose  sum  is  20. 

If  we  have  to  add  2  tons  tons      cwt  quar  lbs>        oz< 

2  cwt.    1  quar.    17  Ibs.    10  3         2  1  17       10 

oz.  avoirdupois,  to  12  tons  12       10  0  2         2 

10  cwt.  2  Ibs.  2  oz.,  2  cwt.  02  1  18         3 

1  quar.    18  Ibs.  3  oz.,  and  Q         Q  Q  911 

9  Ibs.   1 1  oz. ;   then,  from      -^ ^ 3 ^ ^ 

what  was  remarked  above, 

they  will  be  put  down  as  in  the  margin.  Then  the  sum  of 
the  right  hand  column  is  26  oz.,  which  is  1  Ib.  10  oz.,  we 
put  down  the  10  in  the  column  of  oz.,  and  carry  the  1  Ib.  to 
the  column  of  Ibs.  which  is  next ;  and  this  when  added 
comes  to  47  Ibs.,  that  is,  1  quar.  and  19  Ibs. ;  the  19  is  put 
in  the  column  of  Ibs.  and  the  1  is  carried  to  that  of  quars., 
which  comes  to  3,  which  not  amounting  to  1  cwt.  we  put 
down  the  3  in  the  column  of  quars.  and  carry  nothing  to 
the  column  of  cwts.,  which,  when  added,  amounts  to  14,  this 
we  put  down,  and,  as  it  does  not  amount  to  20  cwt.  or  1  ton, 
we  carry  nothing  to  the  column  of  tons  ;  and  when  this  co- 
lumn is  added,  its  sum  is  14. 

25.  In  Subtraction  the  same  principle  of  arrangement  is  to 


28  ARITHMETIC. 

be  observed,  and  the  lesser  quantity  is  to  be  put  under  the 
greater.    If  we  have  to  subtract  1  ton  13  cwt.  2  quars.  17  Ibs 
12  oz.,  from  9  tons  8  cwt.       tons     cwt     quars     lbs        oz 
1  quar.  4  lbs  7  oz.  avoirdu-         98147 
pois,  they  are  arranged  as         j        13         2       17       12 
in  the  margin-     We  begin        ~«       j^ % 14       TT 
to   subtract   at    the   lowest 

denomination,  viz.  oz. — 12  oz.  from  7  oz.  we  cannot,  bu 
we  add  a  Ib.  or  16  oz.  to  the  7,  which  is  supposed  to  be 
borrowed  from  the  column  of  lbs.  which  stands  next  it, 
towards  the  left ;  now  16  added  to  7  makes  23,  and  12  from 
23  leaves  11,  which  is  put  down  .in  the  column  of  oz.  Now 
we  must  pay  back  to  the  column  of  lbs.  the  pound  or  16oz. 
which  we  borrowed,  therefore,  it  is  18  from  4.  Here  we 
have  to  borrow  from  the  column  of  quars.,  and  1  quar. 
being  28  lbs.  we  borrow  28,  then  28  and  4  are  32,  there- 
fore 18  from  32  leaves  T4,  which  is  put  down,  and  the  1 
quar.  paid  back  to  the  column  of  quars.;  3  from  1,  we 
cannot,  and  must  borrow  1  cwt.  or  4  quars.,  therefore  3 
from  5  and  2  remains,  which  is  put  down.  Add  1  to  13  for  the 
1  cwt.  that  was  borrowed,  then  14  from  8,  we  cannot,  but 
borrow  20  from  the  next  column,  then  14  from  28  and  14 
remains.  Pay  back  to  the  column  of  tons  the  1  ton,  or  20 
cwt.  which  we  borrowed,  then  2  from  9  and  7  remains,  which 
is  put  down. 

The  same  principle  holds  in  other  examples,  the  only  va 
riation  being  that  the  numbers  to  be  borrowed  from  the  next 
higher  column,  will  depend  upon  the  relative  values  of  these 
columns,  which  may  be  known  by  examining  a  table  of  the 
particular  weight  or  measure  to  which  the  example  may  refer. 

26.  In  Multiplication,  which  is  only  a  short  way  of  perform- 
ing addition  in  particular  cases ;  the  principles  are  nearly 
similar  :  thus,  to  multiply  3  tons  2  cwt.  2  quars.  6  lbs.  10 
oz.  by  3;  they  are  arranged      tons     cwt      quars       lbs>      oz 
as  in  margin.    1  hen  the  first         3         2         2         6       10 
product  is  30  oz.  or  1  Ib.  3 

which  is  carried  to  the  co-      — Q « ^ r^r rr* 

lumn  of  lbs.,  and  14  oz., 

which  is  put  down  in  the  column  of  oz.  The  product  of 
the  lbs.  is  18,  and  the  one  Ib.  carried  is  19,  which  no 
amounting  to  28  lbs.  or  1  quar.,  nothing  is  to  be  carried  to 
the  column  of  quars.  The  product  of  the  quars.  is  6, 
which  is  1  cwt.  to  be  carried  and  2  quars.  to  be  put  down. 


COMPOUND    NUMBERS.  29 

The  product  of  cwts.  is  6,  and  the  one  carried  from  the 
former  column  makes  7,  nothing  being  carried  ;  the  co- 
lumn of  tons  is  9.  By  examining  the  following  examples, 
and  referring  to  the  tables  of  weights  and  measures,  the 
general  application  may  be  easily  inferred.  See  Appendix 
to  Arithmetic. 

Degrees,      min.     seconds.  yds.        feet.      inch.  8th  parts, 

23          14         17  17         2         9         6 

6  ' 5^ 

139        ^5         42  89       _2       _0       _6 

Carry  by     "60         60  ~3       12       ~8 

27.  It  may  not  be  out  of  place  here  to  notice,  Duodeci- 
mal, or  what  is  commonly  called  Cross  Multiplication ;  which 
is  very  useful  to  artificers  in  general,  in  measuring  timber,  &c. 

The  foot  is  divided  into  12  inches,  each  inch  into  12  parts, 
and  each  part  again  into  12  seconds ;  these  last,  however, 
are  so  small,  that  they  are  generally  neglected  in  calculation. 

If  we  wish  to  find  the  surface  of  a  plank,  whose  breadth 
is   1   foot  7  inches,  and  length  8  feet  5         8         5 
inches,  we  place  the  one  under  the  other,         \         7 

feet  under  feet,  inches  under  inches,  &c.,         Q ^ 

as  in  the  margin.     Multiply  the  inches          .        ,  ~       , . 

and  feet  in  the  upper  line,  by  the  feet 

in  the  under  line,  placing  the  product  ^ 
of  the  inches,  under  the  inches,  and  that  of  the  feet,  under 
the  feet.  Then  multiply  the  inches  and  feet,  of  the  upper 
line,  by  the  inches  in  the  under  line,  placing  the  product 
one  place  further  towards  the  right,  and  carry  by  twelves 
where  necessary ;  as  in  this  example,  7  times  5  is  35,  that 
is,  two  twelves  and  11  over;  the  11  is  put  down,  and  the 
2  added  to  the  product  of  the  next  column, — 7  times  8  is 
56,  and  the  2  carried  makes  58,  that  is  four  twelves  and 
10  over ;  the  10  is  put  down,  and  the  4  carried  to  the  next 
column.  These  are  now  added,  observing  again  to  carry 
by  twelves. 

feet.      inch.  feet.      inch,    parts. 

47  35         4         6 

8         4  12         3         4 

36         8 
164 


38 


424 

6 

0 

8 

10 

1 

6 

11 

9 

6 

0 

434 

3 

11 

0 

0 

3* 

30  ARITHMETIC. 

The  feet  in  the  example  are  square  feet,  but  the  inches 
are  not  square,  as  might  be  thought  at  first  sight,  but  12th 
parts  of  a  square  foot ;  and  also  the  numbers  standing  in 
the  third  place,  are  12th  parts  of  these  12  parts  of  a  foot, 
and  so  on. 

28.  Before  we  consider  the  Division  of  compound  num- 
bers, it  will  be  necessary  to  attend  a  little  to  the  nature  of 
reduction.  This  is  usually  thought  by  beginners  to  be  very- 
perplexing,  but  a  little  attention  to  the  principle,  will  ob- 
viate all  this  apparent  difficulty. 

In  every  lineal  foot  there  are  12  inches,  and  therefore 
there  will  be  12  times  as  many  inches,  in  any  number  of 
feet,  as  there  are  feet;  thus,  in  8  feet  there  are  8  times  12, 
that  is,  96  inches.  In  every  Ib.  avoirdupois  there  are  16 
ounces,  therefore  in  18  Ibs.  there  are  18  times  16,  that  is, 
288  ounces.  So  that  we  multiply  the  higher  denomina- 
tion, by  that  number  of  the  lower  which  makes  one  of  the 
higher,  and  the  product  is  the  number  of  the  lower  contained 
in  the  number  of  the  higher,  which  we  multiply.  In  the  pre- 
vious examples,  feet  and  pounds  are  the  higher  denomina- 
tions, and  inches  and  ounces  are  the  lower.  From  these 
remarks  it  will  be  easy  to  see,  how  we  proceed  in  rinding 
the  number  of  £  parts  of  an  inch  contained  in  3  yards  2 
feet  7  inches,  and  J-  parts,  long  measure.  Bring  the  yards 
to  feet,  3  multiplied  by  3  are  9,  to  which  we  add  the  2 
feet,  which  make  11.  This  brought  to  inches,  is  11  mul- 
tiplied by  12,  or  132,  to  which  we  add  the  7  inches,  making 
139.  This  brought  to  £  parts  gives  139,  multiplied  by  8, 
that  is,  1112,  to  which  we  add  the  5  eighth  parts,  making 
1117  the  answer. 

The  examples  subjoined  are  managed  in  a  like  manner; 
the  multipliers  varying  with  the  kind  of  weight  or  mta- 
eure. 

cwt.        quar.        Ibs.  acres.        roods.        poles. 

27  1         22  22  3  24 

4  mult.  4  mult. 

108  quars.  88  roods 

1  add  3  add 


109  quars.  91  roods 

28  mult.  40  mult. 


3052  Ibs.  3640  poles 

22  add  24  add 

3074  Ibs.  3664  poles. 


COMPOUND    NUMBERS.  31 

The  work  is  reversed,  when  we  wish  to  ascertain  how 
many  of  a  higher  denomination  are  contained  in  any  num- 
ber of  a  lower.  Thus,  in  1440  inches,  long  measure,  there 
will  be  one  foot  for  every  12  inches,  we  therefore  divide 
1440  by  12,  and  the  quotient  will  be  the  number  of  feet, 
that  is,  120  feet.  Then  there  is  no  remainder,  but  if  there 
had,  it  would  have  been  of  the  same  kind  with  the  dividend, 
that  is,  inches.  In  the  same  way  lind  how  many  tons,  cwts. 
quars.  and  Ibs.,  are  contained  in  12345678  oz. 

oz.  in  1  lb.— 16        )  12345678         ounces. 
Ibs.  in  1  quar.— 28    ~)771604  Ibs.— 14  oz. 
quars.  in  cwt. — 4  )27557  quars. — 8  Ibs. 

cwt.  in  1  ton — 20  )6889  cwt. — 1  quar. 

344  tons — 9  cwt. 

The  answer  therefore  is  344  tons  9  cwt.  1  quar.  8  lb.  14 
oz. — which  may  be  proved  by  reducing  the  work  to  ouncef 
by  the  method  given  above. 

29.  It  is  frequently  of  great  use,  to  express  compound 
numbers  fractionally  ;  thus,  so  many  feet  and  inches  as  the 
fraction  of  a  yard.  What  fraction  of  a  yard  is  2  feet  8 
inches  ?  Now,  from  what  has  been  said  on  vulgar  fractions, 
it  will  be  easily  seen  that  one  yard  is  here  the  unit,  or  de- 
nominator of  the  fraction,  which  must  of  course  be  brought 
to  inches.  Now  there  are  36  inches  in  one  yard,  which 
must  be  the  denominator  of  the  fraction,  and  the  numera- 
tor will  be  the  quantity  taken ;  that  is,  2  feet  8  inches  re- 
duced to  inches,  or  32  inches.  The  fraction  therefore  is 
ff,  or  simplilied  £,  which,  turned  into  a  decimal,  is  0*8888, 
one  yard  being  1.  So  likewise,  what  fraction  of  a  cwt.  is 
2  qrs.  14  Ibs.  3  oz.?  This  last  reduced  to  ounces  is  1123, 
which  is  the  numerator  of  the  fraction,  and  the  denomina- 
tor is  1  cwt.  reduced  to  oz.,  or  1792  oz. ;  the  fraction  is 
therefore  jif  ?,  which  is  expressed  decimally  0-6264.  We 
think  that  these  examples  will  be  sufficient  to  show  the 
mode  of  procedure,  and  it  remains  for  us  to  consider  the 
reverse  of  this;  to  estimate  the  value  of  such  fractions  in 
terms  of  the  weight  or  measure  to  which  they  refer. 

3C.  It  will  be  easily  seen,  that  one-half  of  a  foot  is  twelve 
times  greater  than  one-half  of  an  inch,  or  that  any  given 
part  of  a  foot,  is  a  twelve  times  greater  part  of  an  inch  ;  thus, 
5  of  a  foot  is  y  of  an  inch  ;  so  that  to  bring  any  fraction  of 


32  ARITHMETIC. 

a  foot  to  the  fraction  of  an  inch,  we  have  only  to  multiply  the 
numerator  by  12.  So  likewise  $  of  a  pound  avoirdupois,  is 
*/,  of  an  ounce,  and  ^  of  a  yard  is  -^  of  a  foot,  or  3-j  of  an 
.'nch;  and  if  we  divide  the  numerator  by  the  denominator, 
we  get  in  the  last  example  ^  of  a  yard,  equivalent  to  7y 
inches. 

What  is  the  value  of  5  of  1  cwt.  ?  By  applying  the  fore- 
going principle  it  will  be  found  that  5  of  1  cwt.  is  A  of  a 
quar.,  or  a  28  times  greater  part  of  1  lb.,  that  is  l^-2  ;  that  is 
37  5  Ibs.  —  also  |  of  1  lb.  is  16  times  5  of  an  ounce,  or  y, 
equal  to  5|  ounces. 

31.  It  will  generally  be  found  best  to  express  these  deci- 
mally, thus,  the  last  example  will  be  i  of  a  cwt.  or  0.333 
of  a  cwt.,  or  1.333  of  a  quar.,  or  37.666  of  a  pound.    Thus 
it  appears  that  any  fraction  of  a  cwt.  is  4  times  greater  than 
a  like  fraction  of  a  quarter,  and  any  fraction  of  a  quarter 
is  28  times  greater  than  a  similar  fraction  of  a  pound  ; 
hence,  to  reduce  a  fraction  of  a  higher  to  its  value  in  a 
lower  denomination,  we  multiply  the  numerator  of  the  frac- 
tion, by  that  number  which  expresses  how  many  of  the  lower 
are  contained  in  one  of  the  higher,  while  the  denominator 
remains  unaltered.     On  the  other  hand,  to  bring  a  fraction 
from  a  lower  to  a  higher  denomination,  the  numerator  re- 
mains the  same  ;  but  we  multiply  the  denominator  by  that 
number  which  expresses  how  many  of  the  lower  is  contained 
in  one  of  the  higher.     Thus  i  of  an  inch  is  Jff  of  a  foot,  or 
-j-^g-  of  a  yard  ;  or  expressed  in  decimals  0.3333  of  an  inch, 
or  0.0277  of  a  foot,  or  0.00924  of  a  yard. 

32.  On  a  like  principle  the  value  of  a  decimal  expressing 
weight  or  measure,  may  be  determined,  simply  by  multiply- 
ing the  decimal  by  that  number  of  the  next  lower  denomi 
nation,  which  is  contained  in  one  of  the  higher,  and  cutting 
off  the  proper  number  of  decimals  in  the  product,  —  thus  : 

37689  of  a  cwt. 
4 

1.50756  quarters. 

28 
14.21168  pounds. 


3.38688 
Here  it  will  be  observed,  that  the  integers  or  whole  num 


POWERS    AND    ROOTS.  33 

bers  cut  off  are  not  multiplied,  and  the  value  of  .37689  of 
a  cwt.  is  1  quar.  14  Ibs.  3.386  oz. 

We  will  conclude  this  chapter  on  compound  numbers, 
with  some  remarks  on  Division.  The  same  arrangement  ia 
to  be  observed  here  as  in  addition  ;  the  greater  quantity  be- 
ing towards  the  left  of  the  lesser. 

Let  it  be  required  to  divide  13  yards  2  feet  8  inches  by 
4.  We  say  4  in  13,  3  times  and  1  over,  that  is  one  yard, 
which  must  be  reduced  to  feet,  the  next  lower  denomination; 
that  is  3  feet,  and  the  2  feet  are  five  feet — now  4  in  5,  1  and 
1  over,  which  last  being  a  foot,  must  be  reduced  to  inches  ; 
it  is  therefore  12  inches,  and  the  8  make  20 ;  then  4  in  20, 

times  ;  the  answer  therefore  is  3  yards,  1  foot,  5  inches. 

yards.        feet.        inch.        yards.        feet.        inch. 
3)     16  2  9(5  1  11 

15 
~T 
3  mult. 


3 

2  add 

T 
3 

2 

12  mult 


24 

9  add 


3)  33 
33 


POWERS  AND  ROOTS. 

32.  THE  square  of  any  number  is  the  product  of  that 
number  multiplied  by  itself:  thus,  the  square  of  2  is  4,  the 
square  of  4  is  16,  the  square  of  5  is  25,  <fcc.  The  cube  of 
any  number  is  the  product  of  that  number  multiplied  twice 
by  itself :  thus,  the  cube  of  2  is  8,  the  cube  of  3  is  27,  the 
cube  of  4  is  64,  &c.  On  the  other  hand,  when  we  talk  9s 


34  ARITHMETIC. 

the  square  and  cube  roots  of  any  numbers,  we  mean  such 
numbers  that,  when  squared  or  cubed,  will  produce  these 
numbers  :  thus,  2  is  the  square  root  of  4,  3  is  the  square  root 
of  9,  and  4  is  the  square  root  of  16,  &c.  In  like  manner,  3 
is  the  cube  root  of  27,  4  the  cube  root  of  64,  5  the  cube  root 
of  125,  &c.  The  cube  and  cube  root  are  said  to  be  of  higher 
order  than  the  square  and  square  root ;  and  there  are  higher 
orders  than  these,  with  which  we  shall  not  concern  ourselves, 
as  they  will  not  occur  in  our  calculations.  The  method  of 
raising  any  number  to  the  square  and  cube  powers,  will  be 
sufficiently  obvious  from  what  has  been  said  above  ;  but  the 
method  of  extracting  the  square  and  cube  roots  is  not  by  any 
means  so  easy.  We  shall  give  the  rules  for  the  extraction 
of  these  roots  ;  and  as  they  are  long,  we  would  recommend 
the  beginner  to  compare  carefully  each  step  in  the  example, 
with  that  part  of  the  rule  to  which  it  refers  ;  and  by  doing 
so  attentively,  he  will  find  that  the  greater  part  of  the  diffi- 
culty will  vanish. 

33.  The  rule  for  extracting  the  square  root  is  this  : 

First — Commencing  at  the  unit  figure,  point  off  periods 
of  two  figures  each,  till  all  the  figures  in  the  given  number 
are  exhausted.  The  second  point  will  be  above  hundreds 
in  whole  numbers,  and  hundredths  in  decimals. 

Second — If  the  first  period  towards  the  left  be  a  complete 
square,  then  put  its  square  root  at  the  end  of  the  given  num- 
ber, by  way  of  quotient ;  and  if  the  first  period  is  rtot  a  com- 
plete square,  take  the  square  root  of  the  next  less  square. 

Third — Square  this  root  now  found,  and  subtract  the 
square  from  the  first  period ;  to  the  remainder  annex  the 
next  period  for  a  dividend,  and  for  part  of  a  divisor  double 
the  root  already  obtained. 

Fourth — Try  how  often  this  part  of  the  divisor  now  found 
is  contained  in  the  dividend,  omitting  the  last  figure,  and 
annex  the  quotient  thus  found,  not  only  to  the  root  last 
found,  but  also  to  the  divisor,  last  used. 

Fifth — Then  multiply  and  subtract,  as  in  division  ;  to  the 
remainder  bring  down  the  next  period,  and,  adding  to  the 
divisor  the  figure  of  the  root  last  found,  proceed  as  before 

Sixth — Continue  this  process  till  all  the  figures  in  the 
given  number  have  been  used ;  and  if  any  thing  remain, 
proceed  in  the  same  manner  to  find  decimals — adding  two 
ciphers  to  find  each  figure. 


;.s    AND    ROOTS. 


The  square  root  of  365  is  required. 

305(19-1049 
1 


29 
9 


205 
2(U 


400 


38204 
4 


190000 
152816 


382089 
9 


3718400 
3438801 


382098  I     279599 
'fiie  square  root  of  2  to  six  places  of  decimals  is  required. 

.     2  ( 1-414213 
1 


24 
4 


100 
96 


281 

1 


400 

281 


2824 
4 


11900 
11296 


28282 
2 


60400 
56564 


282841 

1 


383600 
282841 


2828423  j  100759 

34.  The  easiest  rule  for  the  extraction  of  the  cube  root 
is  tliis  : 

By  trials,  take  the  nearest  cube  to  the  given  number, 
whether  it  be  greater  or  less,  and  call  it  the  assumed  cube  : 
thus,  if  29  was  the  given  cube  whose  root  was  to  be  ex- 
tracted, then,  3  times  3  times  3,  or  27,  is  the  nearest  less 
cube,  and  4  times  4  times  4,  or  64,  is  the  nearest  greatest 
cube  ;  27  is  the  nearer  of  the  two,  therefore,  27  is  the  as- 
sumed cube. 

Add  double  the  given  cube  to  the  assumed  cube,  and 
multiply  this  sum  by  the  root  of  the  assumed  cube,  and  this 
product  divided  by  the  given  cube,  added  to  twice  the 


39  ARITHMETIC, 

assumed  rube,  Trill  give  a  quotient  which  will  be  th«  r<e* 
quired  root,  nearly, 

By  using,  in  like  manner,  the  cube  of  the  last  nnswer,  as 
an  assumed  root,  and  proceeding  in  the  same  manner,  we 
will  get  a  second  answer  nearer  the  truth  than  the  first,  aiuS 
*o  on. 

Find  the  cube  root  of  21 035-8. 

If  20  is  assumed,  its  eube  is  8000  ;  if  30,  its  cube  is  27000, 
— the  one  a  great  deal  too  small  and  the  other  too  great :  let  us 
therefore  try  some  number  between  them,  as  27  ;  the  cube 
of  this  is  19683,  which  we  shall  call  the  assumed  cube  ;  then, 
— twice  the  assumed  cube  is  39366 — twice  the  given  cube  i» 
12071-6. 

Therefore,  the  sum  of  the  given  cube  and  twice  the  as- 
sumed cube  is  60401-8,  and  the  sum  of  the  assumed  cube 
and  twice  the  given  cube  is  61754-6. 

Wherefore,  by  the  rule, 

61754-6 

27 

4322822 
1235092 


60401-8)  1667374-2(27-6047 

This  quotient  is  the  root  nearly  ;  and  by  using  27*6047  in 
the  same  way  that  we  used  27,  we  will  get  an  answer  still 
nearer  the  true  root.  For  a  Table  of  Powers  and  Root*, 
see  Grier's  Mech.  Diet. 


THE  SLIDING  RULE. 

35.  We  are  indebted  for  the  invention  of  this  useful  in 
strument  to  Edmond  Gunter.  It  is  a  kind  of  logarithmic 
table,  whose  great  use  is  to  obtain  the  solution  of  arithme- 
tical questions  by  inspection,  in  the  multiplication,  division, 
and  extraction  of  the  roots  of  numbers.  It  consists  of  two 
equal  pieces  of  boxwood,  each  12  inches  long,  joined  toge- 
ther by  a  brass  folding  joint.  In  one  of  those  pieces  there 
is  a  brass  slider.  On  the  face  of  this  instrument,  there  are 
engraven  four  lines,  marked  by  the  letters  A,  B,  O,  and  D  : 
at  the  beginning  of  each  line,  the  lines  A  and  I)  being 


THE    SLIDING    RULE.  37 

marked  on  the  wood  part  of  the  rule,  and  B  and  C  on  the 
brass  slider. 

36.  Before  the  use  of  the  sliding  rule  can  be  explained, 
it  is  necessary  that  a  correct  idea,  should  be  formed  of  the 
method  of  estimating  the  values  of  the  several  divisions  on 
these  lines.     Let  it  be  observed,  then,  that  whatever  value 
is  given  to  the  first  1  from  the  left,  the  numbers  following, 
viz.  2,  3,  4,  5,  Ac.,  will  represent  twice,  thrice,  four  times, 
<fcc.,  that  value.  If  1  is  reckoned  1  or  unity,  then  2,  3,  4,  &c., 
will  just  signify  two,  three,  four,  &c. ;  but  if  1  is  reckoned 
iO,  then  2,  3,  4,  <fcc.,  will  represent  20,  30,  40,  &c.    If  the 
first  1  is  reckoned  100.  then  2,  3,  4,  &c.,  will  represent  200, 
300,  400,  &c.     The  value  of  the  1  in  the  middle  of  the  line 
Is-  always  ten  times  that  of  the  first   1  ;  the  value  of  the 
second  2  is  ten  times  that  of  the  first.  2  :  so  that  if  the  value 
of  the  first  1  be  10,  that  of  the  second   1  will  be  100;    the 
first  2  will  he  20,  and  the  second  2  will  be  200,  <fec.     The 
value  of  these  divisions  being  understood,  we  may  now  at- 
tend to  the  minute  divisions  between  these.     Now,  on  the 
lines  A,  B,  and  C,  there  are  50  small  divisions  betwixt  1  and 

2,  2  and  3,  3  and  4,  &c.  ;  and  it  follows,  from  the  nature  of 
the  larger  divisions,  that  if  the  first  1  be  reckoned   1,  or 
unity,  each  of  these  small  divisions  between  1  and  2,  2  and  3, 
&c.,  will  be  -j'g,  or  '02  ;  and  supposing  still  the  first  1  to  be 
unity,  then  the  small  divisions  from  the  second  1  to  2,  2  to 

3,  <kc.,  will  each  be  ten  times  greater  than  aT'5,  or  P02,  that 
is,  each  of  them  will  be  -^°,  or  j,  or  '2.     In  the  same  way, 
if  the  first  1  represents  100,  the -first  2  will  be  200  ;  if  the 
second  1  be  1000,  the  second  2  will  be  2000,  <fec. ;  and  on 
the  same  principle  as  above  the  small  divisions  or  50th  parts 
will  represent  each  ^  of  100,  or  2,  in  the  *:rst  half,  or  from 
the  first  1  to  2,  2  to  3,  &c.,  and  3'ff  of  1000,  or  20,  in  the  second 
half;  or  from  the  second  1  to  the  second  2,2  to  3,  &c. 

37.  These  divisions  being  understood,  we  may  proceed  to 
show  the  method  of  using  this  rule  in  the  solution  of  arith- 
metical questions. 

38.  To  find  the  product  of  two  numbers  : 

Move  the  slider,  so  that  1  on  B  stands  against  one  of  the 
factors  on  A  ;  then  the  product  will  be  found  on  the  line  A, 
against  the  other  factor  on  the  line  B. 

Thus,  to  find  the  product  of  3  by  8  : 

Set  1  on  B  to  3  on  A  ;  then  against  8  on  B  will  be  found 
the  product  24  on  A. 

4 


38  ARITHMETIC. 

For  trie  product  of  34  by  16  : 

Set  1  on  B  against  16  on  A,  then  look  on  B  for  34,  and 
igainst  it  on  the  line  A  will  be  found  the  product  544. 

39.  To  find  the  quotient  of  two  numbers  : 

This  may  be  done  in  two  ways, — either  set  1  on  the  slider 
B  against  the  divisor  on  A,  then  against  the  dividend  on  A 
the  quotient  will  be  found  on  B.  Or,  set  the  divisor  on  B 
against  1  on  A,  then  the  quotient  will  be  found  on  A  against 
the  dividend  on  B  ;  therefore,  in  general,  it  is  to  be  remem- 
bered, that  the  quotient,  must  always  be  found  on  the  same 
line  on  which  1  was  taken,  and  the  divisor  and  dividend  on 
the  other  line. 

Thus,  to  find  the  quotient  of  96  divided  by  6  : 

Move  the  slider  till  1  on  B  stands  against  6  on  A  ;  then 
the  quotient  16  will  be  found  on  B  against  the  dividend 
96  on  A. 

In  like  manner,  to  find  the  quotient  of  108  divided  by  12, 
we  may  take  the  latter  form  of  the  rule,  thus  : 

Set  12  on  B  against  1  on  A  ;  then  on  the  line  A  will  be 
found  the  quotient  9  agains!  96  on  B. 

40.  To  solve  questions  in  the  rule  of  three  or  simple  pro- 
portion 

Set  the  first  term  on  the  slider  B  to  the  second  on  A ; 
th$i  on  the  line  A  wll  be  found  the  fourth  term,  standing 
against  th«j  third  term  on  B. 

If  4  Ibs   of  brass  cost  36  pence,  what  will  12  Ibs.  cost  ? 

Move  the  slider  so,  that  4  on  B  will  stand  against  12  on 
A  ;  then  against  36  on  B  will  be  found  the  fourth  term  108 
on  A. 

41.  To  extract  the  square  root: 

Move  the  slider  so,  that  the  middle  division  on  C,  which 
is  marked  1,  stands  against  10  on  the  line  D,  then  against  the 
given  number  on  C  the  square  root  will  be  found  on  D. 

It  is  to  be  observed  before  applying  this  rule,  that  if  the 
given  number  consists  of  an  even  number  of  places  of  figures, 
as  two,  four,  six,  &c.,  it  is  to  be  found  on  the  left  hand  part 
of  the  line  C  ;  but  if  it  consists  of  any  odd  number  of  places, 
as  three,  five,  seven,  &c.,  it  is  to  be  found  on  the  right  hand 
side  of  C,  1  being  the  middle  point  of  the  line. 

To  find  the  square  root  of  81  : 

Here  the  number  of  places  are  even,  being  two  ;  therefore, 
the  number  81  is  sought  for  on  the  left  hand  side  of  the 
line  C 


.MAKKS    01'    CO  NTH  ACTION.  39 

Set  1  on  C  against  10  on  D  ;  then  against  81  on  C  will 
be  found  9,  the  square  root  on  D. 

For  the  square  root  of  144  : 

Set  1  on  C  to  10  on  D  ;  then  against  144  on  C  will  be 
found  the  square  root  12  on  D. 

42.  To  rind  the  area  of  a  board  or  plank : 

The  rule  is,  to  multiply  the  length  by  the  breadth,  the 
product  will  be  the  area ;  therefore,  by  the  sliding  rule, 

Set  12  on  P»  against  the  breadth  in  inches  on  A  ;  then  on 
the  line  A  will  be  found  the  surface  in  square  feet,  against 
the  length  in  fe(;t  on  the  line  B. 

To  find  the  area  of  a  phnk  18  inches  broad  and  10  feet 
3  inches  lonjj : 

Move  the  slider  so  that  12  on  B  stands  against  18  on  A  ; 
then  will  10|  on  B  stand  against  15*  on  A,  which  15|  is 
square  feet. 

This  may  be  proved  by  cross  multiplication. 
10     3 
_1 G 

10     3 
5     1     6 


15     4     6 

43.  For  the  solid  content  of  timber. 

The  rule  is  to  multiply  length,  breadth,  and  thickness  all 
together. 

Set  the  length  in  feet  on  C  to  12  on  D  ;  then  on  C  will 
be  found  the  content  in  feet  against  the  square  root  of  the 
product  of  the  depth  and  breadth  in  inches  on  D. 

What  is  the  content  of  a  square  log  of  timber,  the  length 
of  which  is  ten  feet,  and  the  side  of  its  square  base  is  15 
'nches. 

Set  10  on  C  against  12  on  D ;  then  will  15  on  D  stand 
against  the  content  15|  on  C. 

44.  Other  particulars  on  the  measurement  of  timber  wil 
be  given  hereafter,  when  we  come  to  Mensuration. 


MARKS  OF  CONTRACTION. 

45.  WE  earnestly  request  that  particular  attention  be  paid 
to  this  chapter,  not  because  it  is  difficult,  but  because  it  is  of 
the  greatest  importance  to  the  clear  understanding  of  what 


4C  ARITHMETIC. 

follows  in  this  book,  and  contributes  greatly  towards  its 
shortness  and  simplicity. 

46.  When  we  mean  to  say  that  one  thing  is  equal  to  an- 
other, we  use  this  mark  =  thus,  3  added  to  5  =  8,  is  read 
thus,  3  added  to  5  is  equal  to  8. 

47.  But  the  words,  added  to,  may  also  be  represented  by 
the  mark  +  thus,  3  +  5  =  8,  is  read,  3  added  to,  or  plus,  5  is 
equal  to  8. 

48.  So  likewise  the  difference  of  two  numbers  may  be 
represented  by  the  mark — ,  which  is  a  short  way  of  express- 
ing the  word  subtract,  thus,  5 — 3=2,  is  read  from  5  sub- 
tract 3  the  difference  is  equal  to  2  ;  and  thus,  3  +  6 — 2=7 
is  a  short  way  of  writing,  to  3  add  6  and  subtract  2,  the  result 
is  equal  to  7. 

49.  After  the  same  manner  the  mark  x  is  used  instead 
of  the  words  multiply  by,  thus,  3x2  =  6,  is  read  3  multi- 
plied by  2  is  equal  to  6. 

50.  To  show  that  the  operation  of  division  is  to  be  per- 
formed this  mark  is  sometimes  used,  viz.  -=-,  which  is  a  short 
way  of  writing  the  words,  divided  by,  thus,  15-r-3  =  5,  is  read 
15  divided  by  3  is  equal  to  5  :  but  we  will  in  general  place 
the  divisor  below  a  line  with  the  dividend  above  it,  on  the 
principle  stated  in  vulgar  fractions,  thus, y  =5  the  same 
as  15-7-3=5. 

51.  The  square  of  any  number  or  quantity  is  marked  by  a 
small  a  placed  at  its  upper  right  hand  corner,  thus,  38=9  is 
read,  the  square  of  3  is  9.    The  cube  is  marked  by  a 3  placed 
in  the  same  way,  as  33=27,  that  is,  the  cube  of  3  is  27. 

The  square  root  is  noted  in  a  similar  manner  by  the  frac- 
tion ^  placed  in  the  same  way,  as  9^  =  3,  and  so  likewise  the 
cube  root,  as  27*  =  3  ;  but  the  square  root  is  often  denoted 
byv/placed  before  the  number  or  quantity,  thus,  v/9  =  9*=3, 
and  the  cube  root,  in  like  manner,  by  <$f,  thus,  v!/27  =27^=3. 

52.  Parenthes'es  ( )  are  used  to  show  that  all  the  numbers 
within  them  are  to  be  operated  upon  as  if  they  were  only  one ; 
thus,  3  +  2x5,  means  that  3  is  to  be  added  to  the  product  of 
2  and  5,  that  is,  the  amount  of  this  is  13 ;  but  (3  +  2)  x5, 
means  that  3  and  2,  that  is,  5,  is  to  be  multiplied  by  5,  and 
the  result  will  be  25  ;  a  very  different  thing  from  what  it 
was  before,  which  arises  entirely  from  the  use  of  paren- 
theses.    In  like  manner  3  +  23=7,  but  (3+2)2=25  ;  here 
as  in  every  other  case,  the  whole  of  the  numbers  wittnn 
the  parentheses  are  taken  as  one  whole,  and  as  such  *»•* 


.  41 

affccted  by  whatever  is  without  the  paretttUese*.  The 
same  thing  is  often  marked  by  drawing  a  line  over  all  the 
numbers  or  quantities  to  be  taken  as  one  whole  ;  thus,  instead 
of  (3-f2)x5,  we  may  write  3-f2x5;  also  (6x4)  —  3x2 
is  the  same  as  6x4  —  3  X  2,  both  being  equal  to  42. 

f>3.  The  rule  for  tin;  measurement  of  the  surface  of  timber, 
given  in  our  remarks  on  the  sliding  ruic,  may  be  expressed 
thus,  length  xbreadtk  s  area;  and  the  rule  for  simple  pro- 
portion, to  be  given  in  the  next  chapter,  may  also  be  writtea 
thus  : 

Second  term  X  third  term, 

—  =iourth  term. 
first  tenu, 

54.  It  is  obvious  that  this  is  merely  a  kind  of  short  hand 
which  might  he  carried  still  farther  ;  for  instance,  in  the 
last  example  we  rnijrla  make  F  stand  for  the  first  term, 
S  for  the  second,  T  for  the  third,  and  £  for  the  last,  aad 
the  rule  woul'i  then  bo 


55.  We  again  insist  that  the  young  reader  will  read  this 
chapter  carei'uiiy  over. 


PROPORTION. 

56.  When  four  numbers  following  eaoh  other  are  such 
that  the  first  is  as  many  times  greater  or  less  than  the 
second,  as  the  third  is  greater  or  less  than  the  fourth,  they 
are  said  to  be  in  proportion;  thus,  2,  4,  3,  6,  usually  written 
thus,  2  :  4  : :  3  :  6  ;  the  taark  :  being  put  for  the  words,  is 
to,  and  : :  for,  as,  so  that  this  would  he  read,  "2  is  to  4  as  8 
is  to  6.  Here  the  first  is  half  the  second,  and  the  third  is 
half  the  fowtVi,  and  they  are  therefore  in  proportion;  bu< 
they  may  be  arranged  otherwise  ant!  yet  he  in  proportion, 
thus,  4  :  2  : :  6  :  •*.  where  the  first  is  twice  as  large  as  the 
second,  and  the  third  is  twice  as  lane  as  tfce  fourth.  In  ali 
cnses  the  two  middle  terms  are  called  the  means,  and  the 
two  oster  terms  are  called  fhe  extremes.  The  product  of 
"the  two  means  is  equal  to  that  of  the  two  extremes,  thus  i» 
the  last  example,  2x6  =  1-2,  and  4x3  =  12.  Now,  if  w« 
wanted  ttae  last  terra,  to  wit,  3,  it  couid  easily  he  &ead  by 


42 

means  of  this  property  of  numbers  in  proportion.  If  we  h;*4 
only  three  terms  given,  ns  4  :  2  • :  6,  to  find  the  fourth  in 
proportion,  which  is  the  last  extreme,  and  4  is  the  nrst 
extreme.  Now,  we  must  lind  $u«h  a  number,  that,  when 
multiplied  by  4,  the  product  will  be  equal  to  the  product  of 
the  means ;  2x6=12,  to  find  such  a  number  we  have  only, 
by  the  definition  of  division,  to  divide  the  product  of  the 
two  means,  viz.  12  by  the  first  extreme  4,  and  the  quotient 
3  will  be  the  answer.  So  universally  6:9::  12:  where 
the  last  term  will  be  found,  as  before,  by  multiply  ing-  the 
two  means  12x9=108,  and  dividing  the  product  108  by 
the  first  extreme  6,  the  quotient  will  be  the  last  extreme  18, 
hence  6:9::  12,:  18.  The  rute  may  be  expressed  simply 
thus:  let  F  stand  for  the  first  term,  S  the  second,  T  the 

third,  and  £  the  last,  then  we  have — =  £,  and  this 

r 

rule  holds  true  whether  the  numbers  be  whole  or  fractional ; 
and  here  it  may  be  observed,  that  it  will  in  most,  if  not  i» 
all  cases,  be  best  to  turn  all  vulgar  fractions,  when  they 
occur,  into  deciaials  ;  thus,  2£  :  3§  : :  6-1  :  or  f  :  V  : :  V  : 

2*  =  -    

31  =  y  =  3-666   ^  2-5  :  3-666  : :  S'25  : 


>i  5 O.^  "1 

"1  —  2      —43 

)|  =  U  =  3-666   I 
%  =  «  —  6-25    J 


Here  the  mode  of  determining-  the  fourth  term  is  the 
same  in  all;  the  two  means  being  x,  and  their  product  --, 
by  the  first  term.  This  is  usually  called  the  rule  of  three, 
and  is  of  the  utmost  utility  in  practical  arithmetic.  We 
shall  now  show  how  it  is  to  be  applied. 

If  we  pay  40  pence  for  2  feet  of  wood,  how  much  will  we 
pay  for  6  feet  at  the  same  rate  ?  Here  it  is  clear  we  will 
pay  in  proportion  to  the  quantity  of  wood  ;  for  as  many 
times  as  we  have  2  feet,  \ve  will  pay  so  many  times  40  pence  ; 
that  is,  the  price  will  be  in  proportion  to  the  quantity  of 
wood.  So  that  we  may  say,,  as  the  one  quantity  of  wood  is 
to  another  quantity,  so  will  be  the  price  of  the  first  quantity 
to  the  price  of  the  second.  Hence  the  terms  in  the  question; 
will  stand  arranged  thus  :  —  2  :  6  ::  40  :  120,  which  term  120 
is  the  price  oi'  6  feet,  and  is  found  by  the  rule  given  above  t 


57.  In  every  question  in  simple  proportion,  there  wiB 
always  be  thsee  terras,  one  of  *vbtth  i»o£  tlve  same  kind 


COMI'Ut  Mi)    I'KOI'OUTION.  4J 

with  the  answer  sought,  whether  it  be  money,  measure, 
time,  force,  or  any  thing,  which  term  in  the  question  we 
put  in  the  third  place  ;  :i<  in  the  h^t  question  the  answer 
w:is  to  be  money,  and  therefore  the  money  in  the  question, 
40  pence,  was  placed  as  the  third  term.  When  this  is  done, 
we  next  consider  whether  the  answer  will  be  greater  or  less 
than  the  third  term,  and  place  the  greater  or  less  of  the 
other  two  terms  next  it  in  the  second  place,  and  the  other 
one  first,  as  the  answer  may  require  ;  after  which,  employ 
the  rule  given  above  to  find  the  answer. 

58.  As,  for  ex'ample,  40  men  will'do  a  piece  of  work  in 
15  days,  in  how  many  days  will  20  men  do  the  same? 
Here  the  answer  must  be  days  ;  consequently,  15  goes  in 
the  third  term,  and  20  men  will  take  more  time  than  40  to 
do  it,  therefore  we  must  put  the  greatest  in  the  second  place, 
and  the  least  in  the  first ;  and  it  therefore  stands  thus : — 
20  :  40  : :  15  :  the  answer  30,  which  is  found  by  the  rule. 

40X15  ^30. 


COMPOUND  PROPORTION. 

51  COMPOUND  PROPORTION  depends  entirely  on  the  sam« 
principles  as  simple  proportion.  For  instance,  if  2  feet  of 
fir  cost  40  pence,  what  will  G  feet  of  mahogany  cost,  3  feet 
of  mahogany  being  equal  in  value  to  9  of  fir.  Here  we 
may  find  the  price  of  the  6  feet  of  mahogany  as  if  they 
were  fir,  and  it  comes  out,  by  the  last  article,  120  pence, 
but  3  is  to  9  as  the  price  of  fir  is  to  that  of  mahogany ; 
therefore  we  put  the  120,  the  price  of  6  feet  of  fir,  in  the 
third  term,  and  state  the  proportion,  3:9::  120  :  360,  the 
price  of  6  feet  of  mahogany.  The  same  would  have  been 
more  easily  found  by  stating  it  thus  : 


6  :  54  : :       40  :  360.  Ans. 

where  the  proportion/  are  stated  under  each  other,  and 
multiplied  together,  which  produces  3x2  =  6  and  6x9 
=  54,  two  terms  of  a  new  proportion,  in  the  simple  rule, 
where  4(  is  the  third  term  ;  and  this  is  only  the  particular 
example  of  a  general  rule,  where  we  may  have  as  many 


44  ARITHMETIC. 

proportions  as  we  please  reduced  to  the  form  of  a  simple 
question  in  the  rule  of  three.  As,  therefore,  that  quantity 
which  is  of  the  same  kind  with  the  required  answer  is  put 
in  the  third  term,  the  rest  will  be  found  to  go  in  pairs  ; 
two  expressing  relation  of  price,  two  relation  of  quality, 
two  relation  of  time,  which  must  be  put  in  proper  order  in 
the  first  and  second  terms,  as  directed  for  simple  propor- 
tion. When  this  is  done,  all  the  first  terms  of  these  several 
proportions  are  to  be  multiplied  together  for  a  new  first 
erm,  all  the  second  terms  together  for  a  new  second  term, 
which  being  placed  with  the  third,  in  the  form  of  simple 
proportion,  and  operated  upon  as  there  directed,  will  give 
the  answer. 

Forty  boys  are  set  to  dig  a  trench  in  summer ;  14  spade- 
fuls can  be  dug  in  summer  for  12  in  winter;  6  men  can  do 
as  much  as  13  boys  ;  and  16  men  can  do  it  in  104  days  in 
winter :  how  long  will  the  boys  take  ?  Here  the  answer 
is  to  be,  how  many  days?  We  have  in  the  question  104 
days ;  the  third  term,  relative  of  difficulty,  14  spadefuls 
and  12  spadefuls  ;  of  strength,  6  men  to  13  boys  ;  relation 
of  numbers,  16  to  40  ;  which  will  be  stated  thus  : 
Relation  of  number,  40  :  16")  makes  the  timeless. 

Relation  of  difficulty,  14  :  12  y ::  104  makes  the  time  less. 
Relation  of  strength,    6:13J  makes  the  time  greater 

Product, 3360  :  2496  : :  104  :  77TVF  days,  Ans. 


ARITHMETICAL  AND  GEOMETRICAL  PROPORTIONS 
AND  PROGRESSIONS. 

60.  THE  subject  of  this  chapter  is  often  referred  to  in 
elementary  books  on  mechanical  science ;  and  for  this  rea- 
son, we  shall  draw  the  attention  of  the  reader,  for  a  little 
while,  to  the  subject. 

61.  When  we  inquire  as  to  the  difference  of  two  num- 
bers, we  inquire  for  their  arithmetical  ratio ;  but  when  we 
inquire  as  to  the  quotient  of  two  numbers,  we  inquire  for 
their  geometrical  ratio.     Thus,  12  —  3  =  9  and  12  -j-  3  = 
4 ;  here  9  is  the  arithmetical  ratio  of  12  and  3,  and  4  is  the 
geometrical  ratio  of  the  same  numbers.     From  this  it  will 
be  seen,  that  ratio  and  relation  are  terms  which  have  the 
same  signification. 


PROPORTIONS    AND    PROGRESSIONS.  45 

62.  When  four  numbers  follow  each  other,  and  are  such 
that  the  difference  of  the  first  two  is  the  same  as,  or  equal 
lo,  the  difference  of  the  last  two,  these  numbers  are  said  to 
be  in  arithmetical  proportion;  thus  the  numbers  12,  7,  9,  4, 
form  an  arithmetical  proportion,  because  the  difference  of 
12  and  7  is  the  same  as  the  difference  of  9  and  4,  both  being 
5.     The  numbers  in  an   arithmetical   proportion   may  be 
varied  in  their  position,  but  still  the  result  will  be  an  arith- 
metical proportion  ;  for  instance,  12,  7,  9,  4,  may  h.e  written 
12,  9,  7,  4,  or  9,  12,  4,  7 :  but  the  most  remarkable  pro- 
perty of  arithmetical  proportion  is  this,  that  the  sum  of  the 
first  and  last  terms  is  always  equal  to  the  sum  of  the  second 
and  third  ;  thus,  12  +  4  =  lt>  and  9  +  7  =  16;  and  from 
this  it  evidently  follows,  that  to  find  the  fourth  term,  we 
add  the  second  and  third  terms  together,"  and  from  their 
sum  subtract  the  first;  the  remainder  is  the  fourth  term. 

63.  An  arithmetical  progression  is  a  series  of  numbers 
such,  that,  in  taking  any  three  numbers  in  succession,  the 
difference  of  the  first  and  second  is  the  same  as  the  differ- 
ence of  the  second  and  third  ;  thus,  1,  2,  3,  4,  5,  6,  7,  8,  or 
14,  12,  10,  8,  6,  4,  2,  where  the  difference  of  the  succeeding 
numbers  in  the  first  is  1,  and  in  the  second  2.   As  the  num- 
bers in  the  first  increase  from  the  beginning,  it  is  called  an 
increasing  arithmetical  series,  or  progression,  and  as  they 
decrease,  from  the  beginning,  in  the  second  example,  it  is 
called  a  decreasing  arithmetical  progression,  or  series. 

64.  Let  us  place  any  one  of  these  progressions  above 
itself,  in  this  manner :— 


2 

14 

4 

12 

6 

8 
8 

10 
6 

12 
4 

14 
2 

16       16       16       16       16       16       16 

writing  the  same  progression  as  increasing  and  decreasing1 
the  respective  terms  of  the  one  being  directly  under  the  re 
spective  terms  of  the  other  in  columns,  as  above,  the  lowest 
line  of  the  three  being  the  sums  of  the  several  columns, 
which  are  all  seen  to  be  16.  Now,  it  will  be  obvious,  that 
the  first  column  consists  of  the  first  and  last  terms  of  the 
series,  2,  4,  6,  <fec.,  with  their  sum,  which  is  16;  the  second 
column  consists  of  the  first  but  one  and  the  last  but  one  of 
the  terms  of  the  same  series,  together  with  their  sum,  which 
is  likewise  16.  The  third  column  consists  of  the  first  but 
two  and  the  last  but  two  terms,  with  their  sum,  which  agaic 


16  AK 

is  16.  We  may  therefore  infer,  that,  in  an  arithmetical 
progression,  the  sum  of  any  two  terms,  equally  distant  from 
the  first  and  last,  is  equal  to  the  sum  of  any  other  two  terms 
which  are  equalry  distant  from  the  first  and  last,  or  equal  to 
the  sum  of  the  first  and  last.  It  will  also  be  seen,  that  the 
under  line,  or  sum  of  the  two  series,  is  therefore  equal  to 
twice  the  sum  of  one  of  the  progressions.  Now,  there  are 
seven  sixteens,  or  112,  which  is  twice  the  sum  of  the  pro- 
gression^therefore  2)112(56  is  the  sum  of  the  progression. 

65.  It  is  also  apparent,  that  if  any  term  be  wanting,  that 
term  may  be  found  by  adding  the  common  difference,  or 
arithmetical  ratio,  of  the  progression,  to  the  term  going  be- 
fore the  term  sought,  or  subtracting  it  from  the  term  which 
follows,  if  the  series  is  increasing,  but  the  reverse  if  decreas- 
ing. Thus,  2,  4,  8,  the  term  awanting  between  the  4  and  8, 
may  be  supplied,  either  by  adding  the  common  difference, 
2,  to  the  4,  or  subtracting  it  from  the  8,  and  we  thus  get  6. 
The  same  may  be  found  by  taking  the  sum  of 'the  terms  on 
each  side  of  the  term  sought,  and  dividing  by  2  ;  thus,  4  -(-  8 
=  12,  then  2)12(6,  the  same  as  before ;  so,  likewise,  3,  5, 
7,  9,  13.     To  fill  up  the  term  awanting  between  9  and  13, 
we  have  9  +  13  =  22,  therefore  2)22(11,  which  is  the 
number  sought,  and  it  is  called  the  arithmetical  mean. 

66.  The  quotient  of  two  numbers  is  their  geometrical 
ratio,  and  thus  a  fraction,  as  j\,  expresses  the  ratio  of  6  to 
12,  and  therefore  1  :  2  : :  6  : 12  is  the  same  thing  as  5  =  T^. 
We  thus  get  another  view  of  the  rule  of  three,  and  it  is  use- 
ful to  view  any  subject  of  this  kind  in  different  ways,  as  by 
so  doing  we  acquire  a  more  accurate  and  extensive  know- 
ledge of  its  nature  and  application.    The  limits  of  this  book 
will  not  permit  us  to  dwell  on  this  subject,  as  we  have  dis- 
cussed the  subject  of  proportion  in  a  former  chapter. 

67.  In  a  series  of  progression  of  numbers,  as  2,  4,  8,  16, 
32,  64,  where  the  quotient  of  any  term,  and  that  which  fol- 
•ows  it,  is  equal  to  the  quotient  of  any  other  term,  and  that 
which  follows  it,  such  progression  is  said  to  be  geometrical. 

68.  Let  us  take  the  geometrical  progression,  2,  6,  18,  54., 
162,  and  write  it  as  we  did  the  arithmetical,  both  as  in  :a- 
f reasing  and  decreasing  series,  thus : — 

2  6         18         54       162 

162         54         18  6  2 


324       324       324       324       324 


P.iOI'OIM  ION.-,    AND 


47 


Here  we  observe  that  the  product  of  the  terms  of  each 
column  is  the  same,  whatever  column  we  take;  and  we 
arrive  at  a  knowledge  of  the  fact,  that  the  product  of  the 
first  and  last  terms  is  the  same  as  the  product  of  any  other 
two  terms,  one  of  which  is  as  many  places  distant  from  the 
first  as  the  other  is  dista"*  from  «he  last  term. 

69.  If  one  term  in  t1**.  above  se.'ics  were  wanting,  for  in- 
stance the  second,  that  is.  t»,  take  the  terms  on  each  side  of 
it,  and  find  their  product,  2  x  IB  =  36,  now  the  square  root 
of  this,  or  6,  will  be  the  number  sought,  which  is  called 
the  geometrical  mean.     In  like  manner  we  might  Knd  the 

.etrical  mean  between  18  and  162  ;  thus,  18  x  102  = 
29 IB,  th.e  square  root  of  which  is  2916.|  =  54,  the  number 
sought.  The  geometrical  mean  is  sometimes  called  the 
mean  proportional. 

70.  The  sum  of  any  geometrical  series  may  be  found 
thus  : 

i  The  greater  extreme  X  ratio) —  less  extreme, 

— -•=  the  sum  of  series, 
ratio  —  1 

thus  the  sum  of  the  last  series  is — 
(162x3)— 2        480  —  2        484 

3-1 =  '  -T-  =1T  =  242' the  sum' 

71.  Terms  relating  to  proportion  often  occur  in  books 
read  by  mechanics,  of  which  it  would  be  useful  to  know 
the  signification;  and,  to  prevent  their  being  misapplied, 
we  give  the  following  illustration.     If  there  be  four  num- 
bers in  proportion,  as  4  :  16  : 

Directly, 4 

Alternately, 4 

Inversely 16 

C  Compounded,.. .4  -f-  16 

{That  is 20 

5  Divided, 4  — 

{That  is 12 

("Con  verted,.. ..4 

{That  is, 4 
Also, 4 
That  is 4 

$  Mixed, 4  4-  16 

{That  is, 20 

To  these  may  be  added,  duplicate  rulio,  or  ratio  of  th 


16: 

3  :  12,  then, 

16 

3           12 

3 

16 

12 

4 

12 

3 

16 

16 

3  4-  12 

12 

16 

15 

12 

-16 

16 

3  —  12 

12 

16 

9 

12 

164-4 

3 

12  4-  3 

20 

3 

15 

16  —  4 

3 

12  —  3 

12 

3 

9 

4  —  16 

34- 

12 

3  —  12 

12 

15 

9 

48  ARITHMETIC. 

squares  ;  triplicate  ratio,  or  ratio  of  the  cubes  ;  sub-dupli- 
cate ratio,  or  ratio  of  the  square  roots  ;  and  sub-triplicate 
ratio,  or  ratio  of  the  cube  roots. 


POSITION. 

72.  Posrnox  is  a  rule  in  which,  from  the  assumption  of 
one  or  more  fatae  answers  to  a  problem,  the  true  one  is 
obtained. 

7H.  It  admits  of  two  varieties,  single  position  and  double 
position. 

74.  In  single  position  the  answer  is  obtained  by  one  as- 
sumption ;  in  double  position  it  is  obtained  by  two. 

75.  Single  position  maybe  applied  in  resolving  problems, 
in  which  the  required  number  is  any  how  increased  or  dimi- 
nished in  any  given  ratio  ;  such  as  when  it  is  increased  or 
diminished  by  any  part  of  itself,  or  when  it  is  multiplied 
or  divided  by  any  number. 

76.  Double  position  is  used,  when  the  result  obtained  by 
increasing  or  diminishing  the  required  number  in  a  given 
ratio,  is  increased  or  diminished  by  some  number  which  ig 
no  known  part  of  the  required  number  ;  or  when  any  root 
or  power  of  the  required  number,  is  either  directly  or  in- 
directly contained  in  the  result  given  in  the  question. 

SINGLE    POSITION. 

77.  Rule. — Assume  any  number,  and  perform  on  it  the 
operations  mentioned  in  the  question  as  being  performed  on 
the  required  number.     Then,  as  the  result  thus  obtained  is 
to  the  assumed  number,  so  is  the  result  given  in  the  ques- 
tion to  the  number  required. 

Exam. — Required  a  number  to  which  if  one  half,  one- 
third,  one-fourth,  and  one-fifth  of  itself  be  added,  the  sum 
may  be  1644. 

Suppose  the  number  to  be  60 :  then,  if  to  60  one- 
half,  one-third,  one-fourth,  and  one-fifth  of  itself  be 
added,  the  sum  is  137.  Hence,  according  to  the  rule,  as 
137  :  1644  :  :  60  :  720,  the  number  required.  The  truth 
of  the  result  is  proved  by  adding  to  720  one-half,  one-third, 
<fec.,  of  itself,  and  the  sum  is  found  to  be  1644.  The  num- 
ber 60  was  here  assumed,  not  as  being  near  the  truth,  but 


POSITION.  49 

•«  being  a  multiple  of  2,  3,  4,  and  5;  and  in  this  way  the 
operation  was  kept  free  from  fractions.  By  the  assumption 
of  any  other  number,  however,  the  answer  would  have 
been  found  correctly,  but  not  so  easily.  The  reason  of  the 
operation  is  so  obvious  as  not  to  require  illustration. 

DOUBLE    POSITION. 

78.  Rule. — Assume  two  different  numbers,  and  perform 
on  them  separately  the  operations  indicated  in  the  question. 
Then,  as  the  difference  of  the  results  thus  obtained  is  to 
the  difference  of  the  assumed  numbers,  so  is  the  difference 
between  the  true  result  and  either  of  the  others  to  the  cor- 
rection to  be  applied  to  the  assumed  number  which  gave 
this  result.     Add  the  correction  to  this  number,  if  the  cor- 
lesponding  result  was  too  small  ;  otherwise,  subtract  it. 

79.  A  more  general  rule  is  this.     Having  assumed  two 
different  numbers,  perform  on  them  separately  the  opera- 
tions indicated  in  the  question,  and  find  the  errors  of  thj 
results.  Then,  as  the  difference  of  the  errors,  if  both  results 
be  too  great  or  both  too  little,  or  as  the  sum  of  the  errors, 
if  one  result  be  too  great  and  the  other  too  small,  is  to  the 
difference  of  the  assumed  numbers,  so  is  either  error  to  the 
corrections  to  be  applied  to  the  number  that  produced  that 
error. 

80.  When  any  root  or  power  of  the  required  number  is 
in  any  way  contained  in  the  result  given  in  the  question, 
the  preceding  rules  will  only  give  an  approximation  to  the 
required  number.    In  this  case  the  assumed  numbers  should 
be  taken  as  near  the  true  answer  as  possible.     Then,  to  ap- 
proximate the  required  number  still  more  nearly,  assume 
for  a  second  operation  the  number  found  by  the  first,  and 
that  one  of  the  two  first  assumptions  which  was  nearer  the 
true  answer,  or  any  other  number  that  may  appear  to  be 
nearer  it  still.     In  this  way,  by  repeating  the  operation  as 
often  as  may  be  necessary,  the  true  result  may  be  approxi- 
mated to  any  assigned  degree  of  accuracy. 

81.  It  may  be  further  observed  also,  that  the  method  of 
double  position,  besides  its  use  in  common  arithmetic,  is  of 
rruch  utility  in  algebra,  affording  in  many  cases  a  very  con- 
venient mode  of  approximating  the  roots  of  equations,  and 
rinding  the  value  of  unknown  quantities  in  very  compli- 
rated  expressions,  without  the  usual  reductions. 

82    Exam.  1. — Required  a  number,  from  which,  if  2  be 
5 


$0  AKITHMKTir 

subtracted,  one-third  of  the  remainder  will  be  5  less  than 
half  the  required  number. 

Here,  suppose  the  required  number  to  be  8,  from  which 
take  2,  and  one-third  of  the  remainder  is  two.  This  being 
taken  from  one-half  of  8,  the  remainder  is  2,  the  first  result. 
Suppose,  again,  the  number  to  be  32,  and  from  it  take  2  : 
one-third  of  the  remainder  is  10,  which  being  taken  from 
the  half  of  32,  the  remainder  is  6,  the  second  result.  Then, 
the  difference  of  the  results  being  4,  the  difference  of  the 
assumed  numbers  24,  and  the  difference  between  5,  the  true 
result,, and  6,  the  result  nearest  it,  being  1  ;  as  4  :  24  :  :  1  :  6, 
the  correction  to  be  subtracted  from  32,  since  the  result  6 
was  Joo  great.  Hence,  the  required  number  is  26. 

83.  Exam.  2. — If  one  person's  age  be  now  only  four 
times  as  great  as  another  person's,  though  7  years  ago  it 
was  six  times  as  great ;  what  is  the  age  of  each  ? 

Here,  suppose  the  age  of  the  younger  to  be  12  years ;  then 
will  the  age  of  the  older  be  48.  Take  7  from  each  of  these, 
and  there  will  remain  5  and  41,  their  ages  7  years  ago. 
Now,  6  times  5  is  30,  which,  taken  from  4i,  leaves  an  error 
of  1 1  years.  By  supposing  the  age  of  the  younger  to  be  1 5, 
and  proceeding  in  a  similar  manner,  the  error  is  found  to 
be  5  years.  Hence,  as  6,  the  difference  of  the  errors,  (both 
results  being  too  small,)  is  to  3,  the  difference  of  the  as- 
sumed numbers,  so  is  5,  the  less  error,  to  2|,  the  correction  ; 
which,  being  added  to  15,  the  sum,  172,  is  the  age  of  the 
younger,  and  consequently  that  of  the  older  must  be  70. 

Both  the  rules  above  given  for  double  position  depend 
on  the  principle,  that  the  differences  between  the  true  and 
the  assumed  numbers,  are  proportional  to  the  differences 
between  the  result  given  in  the  question  and  the  results 
arising  from  the  assumed  numbers.  This  principle  is  quite 
correct  in  relation  to  all  questions  which  in  algebra  would 
be  resolved  by  simple  equations,  but  not  in  relation  to  any 
others  ;  and  hence,  when  applied  to  others,  it  gives  only 
approximations  to  the  true  results. — The  subject  is  of  too 
little  importance  to  claim  further  illustration  in  this  place. 

84.  Exam.  3. — Required  a  number  to  which,  if  twice 
its  square  be  added,  the  sum  will  be  100. 

It  is  easy  to  see  that  this  number  must  be  between  6  and 
7.  These  numbers  being  assumed,  therefore,  the  sum  of  6 
and  twice  its  square  is  78,  and  the  sum  of  7  and  twice  its 
square  is  105.  Then,  as  105—78  :  7—6  • :  105—100  •  -18 


U'KICIirS    AM)     MKAS1IKKS.  51 

which,  beintr  taken  from  7.  the  remainder*  G-82,  is  the 
required  number,  n"-.r!v.  ' 'J'o  this  let  twice  i:.<  square  he 
added,  and  tlie  result  b*90'8448.  Then.  :is  105—99-8448  : 
7 — 6-82  :  :  105 — 100  :  '1740  ;  which,  bein<r  taken  from  7, 
the  remainder  is  (V8254,  the  required  number  still  mor« 
nearly  ;  and  if  the  operation  were  repeated  with  this  and 
the  former  approximate  answer,  the  required  number  would 
be  found  true  for  seven  or  ei^ht  figures. 


APPENDIX    TO    A  R  IT  H  M  E  T  I  C, 

CtlNTAINMNR 

TABLES  OF  WEIGHTS  A.\U  MEASURES. 


ENGLISH. 
AVOIRDUPOIS    WEIGHT. 

Drachms. 

10  =  1  Ounce. 

256  =          16   =          1  Pound. 
7168   =       44H   =       28   =      1  Quarter. 
286782   =     1792   =     1 12   =     4   =      1  Cwt.     . 
573440   =  35840  =  2240   =  80   =  20   =   1  Ton. 
Tons  are  marked  t.  ;    hundred  weights,  r.wt.  ;  quartets, 
yr.  ,•  pounds,  Ib.  ;  ounces,  oz.  ;  and  drachm?,  dr. 

TROY    WEIGHT. 

Grains. 

24  =        I  Pennyweight. 
480   =     20   =      1  Ounce. 
5760   =  240   =   12   =   1  Pound. 

Pounds  are  marked,  lb.;    ounces,  oz. ;     pennyweights, 
dwt. ;  and  grains,  gr. 

LONG    MEASURE. 

Barley  corns. 

'3  =  1  Inch. 

36  =         12  =       1  Foot. 
108  =         36  =       3     =        1  Yard. 
594  =^       198  =      16-5=       5-5=     1  Pole. 
23760  =    7920  =   660     =220    =  40  =  1  Furlong 
190080  =  63360  =5280    =1760    =320  =  8  =  1  Mile, 


SQUARE  MEASURE. 

Inches. 

144  =  1  Foot. 

1296  =  9  =    1  Yard. 

39204  =  272|  =   30j  =   1  P.  le. 

1568160  =  10890  =  1210  =  40  =  1  Rood. 

6272640  =  43560  =  4840  =  160  =  4  =  1  Acre. 

SOLID    MEASURE. 

Inches. 

1728  =     1  Foot. 
46656  =  27   =   1  Yard. 

WINE    MEASURE. 

Pints. 

2  =s  1  Quart. 

8  =  4  =       1  Gallon. 

336  =  168  =    42  =   1  Tierce. 

504  =  257  =    63  =   1-5=  1  Hogshead. 

672  =  336  =    84  =  2    =  1-5=  1  Puncheon. 

1008  =  504  =  126  =   3    =2    =  1-5=  1  Pipe. 

2016  =  1008  =  252  =  6    =4    =3    =2  =  1  Tun 

ALE  AND    BEER  MEASURE. 

Pints. 

2  =      1  Quart. 

8  =      4  =       1  Gallon. 
72  =    36  =      2=1  Firkin. 
144  =    72  =    18  =    2  =  1  Kilderkin. 
288  =  144  =    36  =    4=2  =  1  Barrel. 
432  =  216  =    54  =    6  =  3  =  1-5=  1  Hogshead. 
576  =  288  =    72  =    8=4=2    =  1-5=  1  Puncheon 
864=432  =  108  =  12=6=3    =2    =  1-5=1  Bu  * 

DRY    MEASURE. 

Pints. 

8  =       1  Gallon.' 

16  =      2  =      1  Peck. 

64  =       8  =       4=1  Bushel. 

256  =    32  =     16  =    4  =     1  Coom. 

512  =    64  =    32  =    8   =    2   =     1  Quarter. 

2560  =  320  =  160   =  40  =  10  =     5=1  Wey. 

5120  =  640  =  320  =  80  =  20  =  10  «  2   =  1  Last 


WKIGHTS    AND    MEASURES.  53 

TIME. 

GO  seconds  =  1  minute,     00  minutes  =  1  hour, 

24  hours      =  1  clay,         365]  days      =  1  year,  nearly. 

THE  CIRCLE. 

The  circle  is  divided  into  360  equal  parts,  called  degrees. 
Seconds. 

60   =  1  Minute. 

360   =         60  =        1  Degree. 
32400   =     5400   =     90   =    1  Quadrant. 
129600  =  21600  =  300  =  4  =   1  Circumference. 

Degrees,  minutes,   and  seconds,  are  marked  °,  ',  ";  as, 
4°  5'  6" — 4  degrees,  5  minutes,  6  seconds. 


REMARKS  OX  ENGLISH  WEIGHTS  AND  MEASURES. 

Troy  weight  is  used  frequently  by  chemists,  and  also  in 
weighing  gold,  silver,  and  jewels  ;  but  all  metals,  except 
gold  and  silver,  are  weighed  by  avoirdupois  weight. 

175  troy  pounds  are  equal  to  144  avoirdupois  pounds. 

175  troy  ounces  =  192  avoirdupois  ounces. 

14  oz.,  11  dwt.,  15.J  grs.  troy  =  1  Ib.  avoirdupois. 

18  dwt.,  5.j  gr.  troy  =  1  oz.  avoirdupois. 

3  miles  long  measure  =  1  league. 

69-pV  English  miles  =  60  geographical  miles. 

1089  Scottish  acres  =  1369  English  acres. 

A  chaldron  of  coals  in  London  =  36  bushels,  and  weighs 
3136  Ibs.  avoirdupois,  or  nearly  1  ton,  8  cwt. 

The  ale  gallon  contains  282  cubic  inches,  and  the  wine 
gallon  contains  231  cubic  inches — the  wine  gallon  being  to 
the  ale  gallon  nearly  as  1  Ib.  avoirdupois  to  1  Ib.  troy. 

By  an  Act  of  Parliament  passed  in  1824,  and  carried 
into  execution  in  1826,  imperial  weights  and  measures  were 
introduced  by  this. 

The  pound  troy  contains  5760  grains. 

The  pound  avoirdupois  contains  7000  grains. 

The  imperial  gallon  contains  277'274  cubic  inches. 

The  bushel  (dry  measure)  contains  2218-192  cubic 
inches. 

To  find  the  value  of  the  old  in  terms  of  the  new,  or  the 
reverse,  the  following  table  of  multipliers  is  given. 

5* 


54  AKITHMETIC. 

Dry.  Wine.  Ale. 

To  convert  the  old  into  new  x  0-96943    0-83311     1-01704 
To  convert  new  into  old      x  1-03153    1-20032    0-98324 

Example.fi. — What  is  the  value  in  imperial  measure,  of 
32  wine  gallons  old  measure  ? 

•83311  x  32  =  26-65952  imperial  gallons. 

In  like  manner  4  bushels  imperial  measure  =  1*03153  X 
4  =  4-12612  old  or  Winchester  bushels. 


FRENCH  WEIGHTS  AND  MEASURES. 

Old  System. 

English  Troy  Grains. 
The  Paris  Pound   =    7561 

Ounce  =  472-5625 
Gros  =  59-0703 
Grain  =  -8204 

Eng.  Inches. 
The  Paris  Royal  Foot  of  12  Inches     =     12-7977 

The  Inch =       1-0659 

The  Line,  or  one-twelfth  of  an  inch     =         -0074 

Eng.  Cubical  Feet. 

The  Paris  Cubic  Foot =     1-211273 

The  Cubic  Inch =       -000700 

MEASURE    OF    CAPACITY. 

The  Paris  pint  contains  58-145  English  cubical  inches, 
and  the  English  wine  pint  contains  28-875  cubical  inches ; 
or  the  Paris  pint  contains  2-0171082  English  pints  ;  there- 
fore to  reduce  the  Paris  pint  to  the  English,  multiply  by 
2-0171082. 

New  System. 

MEASURES    OF    LENGTH. 

English  Inches. 

Millimetre  =  -03937 

Centimetre •••••      =  -39370 

Decimetre =         3-93702 

Metre  =       39-37022 

Decametre =     393-70226 


VVKIGHTS  AND 


55 


Hecatometre =  3937-02260 

Chiliometre =  3'J. HO  "22601 

Myriometre =  3P37iW'26014 

M.    P.  Y.     Ft.        In 

A  "Decametre  is =     00  10     2       9-7 

A.  Hecatometre =     0     0  )09     1          '1 

A  Chiliometre =     04  213     1     10-2 

A  Myriometre =     6     1  156     0         -0 

Eight  Chiliometres  are  nearly  5  English  miles. 

MEASURES    OF    CAPACITY. 

English  Cubic  Inches. 

Millilitre =  -06102 

Centilitre =  -61024 

Decilitre =  6-10244 

Litre =  61-02442 

Decalitre =  610-24429 

Hecatolitre =  6102-44238 

Chiliolitre =  61024-42878 

Myriolitre =  610244-28778 

A  Litre  is  nearly  2g  wine  pints. 

14  Decilitres  are  nearly  3  wine  pints. 

A  Chiliolitre  is  a  tun,  12-75  wine  gallons. 


WEIGHTS. 

English  Grains. 

=  -0154 

=  -1544 

=  1-5444 

=  15-4440 

=  154-4402 

=  1544-4023 

=  15441-0234 

==  154440-2344 

A  Decagramme  is  6  dwts.   10-44  gr.  troy ;  or  5-65  dr 
Avoirdupois. 

A  Hecatogramme  is  3  oz.  8-5  dr.  avoirdupois. 
A  Chiliogramme  is  2  Ibs.  3  oz.  5  dr.  avoirdupois. 
A  Myriogramme  is  22 — 1'15  oz.  avoirdupois. 
100  Myriogrammes  are  1  ton,  wanting  32-8  Ibs. 


Milligramme-  •••• 

Centigramme 

Decigramme • 

Gramme • •• 

Decagramme 

Hecatogramme 

Chiliogramme  (Kilogram)- 
Mvriofframme 


56  ARITHMETIC. 

AGRARIAN    MEASURES 

Are,  .  square  Decametre —      =     3-95  Perches. 

Hecatare =     2   Acres,  1  Rood,  30'1 

Perches. 


FIR  WOOD. 

Decistre,  l-10th  Stere =     3-5315  cub.  ft.  Erg 

Stere,  1  Cubic  Metre =  35-3150  cub.  ft. 

DIVISION    OF    THE    CIRCLE. 

100  seconds  =  1  minute. 

100  minutes  =  1  degree. 

100  degrees  =  1  quadrant. 

4  quadrants  =  1  circle. 

THE    ENGLISH    DIVISION. 

60  seconds      =     1  minute. 
60  minutes      =      1  degree. 
360  degrees       =     1  circle. 

DIMENSIONS  OF  DRAWING  PAPER  IN  FEET  AND  INCHES. 

Ft.    In.  Ft.  In. 

Demy   1     7£    X      1     3£ 

Medium  1    10      X      16 

Royal 20x17 

Super  royal  23X17 

Imperial  25X19* 

Elephant 2     3|    X      1    10^ 

Columbier 2     9|    X      1   11 

Atlas 29x22 

Double  elephant 34x22 

Wove  antiquarian 44x27 


GEOMETRY. 


DEFINITIONS. 

1.  A  POINT  is  (hat  which  has  position, 
but  no  magnkude,  nor  dimensions  ;  neither 
length,  breadth,  nor  thickness. 

2.  A  Line  is  length,  without  breadth  or 
thickness. 

3.  A  Surface  or  Superficies,  is  an  exten- 
sion or  a  figure  of  two  dimensions,  length 
and  breadth ;  'out  without  thickness. 

4.  A  Body  or  Solid,  is  a  figure  of  three 
dimensions,  namely,  length,  breadth,  and 
depth  or  thickness. 

5.  Lines  are  either  Right,  or  Curved,  or  mixed  of  these 
two.  ^ 

6.  A  Right  Line,  or  Straight  Line,  lies  all  in  the  same 
direction,  between  its  extremities  ;  and  is  the  shortest  dis- 
tance between  two  points. 

When  aline  is  mentioned  simply,  it  means  a  Right  Lins 

7.  A  Curve  continually  changes  its  di- 
rection between  its  extreme  points. 

8.  Lines  are  either  Parallel,  Oblique, 
Perpendicular,  or  Tangential. 

9.  Parallel  lines  are  always  at  the  same 
perpendicular  distance  4    and    they  never 
meet,  though  ever  so  far  produced. 

10.  Gbliqr.e  lines  change  their  distance, 
and  would  meet,  if  produced  on  the  side 
of  the  least  distance. 

11.  One  line  is  Perpendicular  to   an- 
other, w'lien  it  inclines  not  more  on  the 
•one  side  than  the  other,  or  when  the  angles 
on  both  sides  of  it  are  equal. 

12.  A  line  or  circle  is  Tangential,  or 
;is  a  t:m<retit    to  a   circle  or  other  curve, 
when  it  touches  it  without  cutting,  whea 
»oth  xre 

f 


58 


ARITHMETIC. 


13.  An  Angle  is  the  inclination  or  open- 
ing of  two  lines,  having  different  direc- 
tions, and  meeting  in  a  point. 

14.  Angles  are  Right  or  Oblique,  Acute  or  Obtuse. 

15.  A  Right  Angle  is  that  which  is  made 
by  one  line  perpendicular  to  another.    Or 
when  the  angles  on  each  side  are  equal  to 
one  another,  they  are  right  angles. 

16.  An  Oblique  Angle  is  that  which  is 
made  by  two  oblique  lines  ;  and  is  either 
less  or  greater  than  a  right  angle. 

17.  An%A.cute  Angle  is  less  than  a  right 
angle. 

18.  An  Obtuse  Angle  is  greater  than  a 
right  angle. 

19.  Superficies  are  either  Plane  or  Curved. 

20.  A  Plane  Superficies,  or  a  Plane,  is  that  with  which 
a  right  line  may,  every  way,  coincide.     Or,  if  the  line 
touch  the  plane  in  two  points,  it  will  touch  it  in  every  point. 
But,  if  not,  it  is  curved. 

21.  Plane  Figures  are  bollhded  either  by  right  lines  or 
curves. 

22.  Plane  figures  that  are  bounded  by  right  lines  have 
names  according  to  the  number  of  their  sides,  or  of  their 
angles  ;  for  they  have  as  many  sides  as  angles ;  the  lea?t 
number  being  three. 

23.  A  figure  of  three  sides  and  angles  is  called  a  Tri- 
angle.    And  it  receives  particular  denominations  from  the 
relations  of  its  sides  and  angles. 

24.  An  Equilateral   Triangle   is    that 
whose  three  sides  are  all  equal. 

25.  An  Isosceles  Triangle  is  that  which 
has  two  sides  equal. 


26.    A  Scalene  Triangle  is  that  whose 
three  sides  are  all  unequal. 


27.    A   Right-angled   Triangle   is   that 
which  has  one  right  angle. 


DKFINITIOJfS. 


28.  Other  triangles  are  Oblique-angled, 
and  are  eilher  obtuse  or  acute. 

29.  An  Obtuse-Angled  Triangle  has  one 
obtuse  aniilf. 

30.  An  Acute-angled  Triangle  has  all 
its  three  angles  acute. 

31.  A  figure  of  four  sides  and  angles  is  called  a  Quad- 
rangle, or  a  Quadrilateral. 

32.  A  Parallelogram  is  a  quadrilateral  which  has  both 
its  pairs  of  opposite  sides  parallel.     And  it  takes  the  fol- 
lowing particular  names,  viz.  Rectangle,  Square,  Rhombus, 
Rhomboid. 

33.  A  Rectangle   is    a   parallelogram, 
having  right  angles. 

34.  A  Square   is    an   equilateral  rec- 
tangle ;    having   its   length   and    breadth 
equal. 


35.  A  Rhomboid  is  an  oblique-angled 
parallelogram. 

36  A  Rhombus  is  an  equilateral  rhom- 
boid ;  having  4all  its  sides  equal,  but  its 
angles  oblique. 

37.  A    Trapezium    is    a  quadrilateral 
which  hath  not  its  opposite  sides  parallel. 

38.  A  Trapezoid  has  only  one  pair  of 
opposite  sides  parallel. 

39.  A  Diagonal  is  a  line  joining  any 
two  opposite  angles  of  a  quadrilateral. 

40.  Plane  figures  that  have  more  than  four  sides  are,  in 
general,  called  Polygons  ;  and  they  receive  other  particular 
names,  according  to  the  number  of  their  sides  or  angles. 
Thus, 

41.  A  Pentagon  is  a  polygon  of  five  sides  ;  a  Hexagon 
of  six  sides;   a  Heptagon,  seven;    an  Octagon,  eight;    a 
Nonagon,  nine  ;  a  Decagon,  ten  ;  an  Undecagon,  eleven  ; 
and  a  Dodecagon,  twelve  sides. 

42.  A   Regular  Polygon  has   all  its   sides  and  all  its 
angles  equal.  —  If  they  are  not  both  equal,  the  polygon  is 


43.  An  Eauikteral  Triangle  is  also  a  regular  figure  of 


60 


GEOMETRY 


Ihree  sides,  and  the  Square  is  one  of  four :  the  former  being 
also  called  a  Trigon,  and  the  latter  a  Tetragon. 

44.  Any  figure  is  equilateral,  when   all   its  sides   are 
equal :  and  it  is  equiangular  when  all  its  angles  are  equal. 
When  both  these  are  equal,  it  is  a  regular  figure. 

45.  A  Circle  is  a  plane  figure  bounded 
by  a  curve  line,  called  the  Circumference, 
which  is  everywhere  equidistant  from  a 
certain  point  within,  called  its  Centre. 

The  circumference  itself  is  often  called  a  Circle,  and 
also  the  Periphery. 


46.  The  Radius  of  a  circle  is  aline  drawn 
from  the  centre  to  the  circumference. 


47.  The  Diameter  of  a  circle  is  a  line 
drawn  through  the  centre,  and  terminating 
at  the  circumference  on  both  sides. 

48.  An  Arc  of  a  circle  is  any  part  of  the 
circumference. 

49.  A  Chord  is  a  right  line  joining  the 
extremities  of  an  sfrc. 


50.  A  Segment  is  any  part  of  a  circle 
hounded  by  an  arc  and  its  chord. 

51.  A  Semicircle  is  half  the  circle,  or  a 
segment  cut  off  by  a  diameter. 

The  half  circumference   is  sometimes 
called  the  Semicircle. 

52.  A  Sector  is  any  part  of  a  circle 
which  is  bounded  by  an  arc,  and  two  radii 
drawn  to  its  extremities. 

53.  A  Quadrant,  or  Quarter  of  a  circle, 
is  a  sector  having  a  quarter  of  the  circum- 
ference for  its  arc,  and  its  two  radii  are 
perpendicular  to  each  other.     A  quarter 
of  the  circumference  is  sometimes  called 
a  Quadrant. 


A 

0 

as 


DEFINITIONS. 

54  The  Height  or  Altitude  of  a  figure 
is  a  perpendicular  let  fall  from  an  angle, 
or  its  vertex,  to  the  opposite  side,  called 
the  base. 

55.  In  a  right-angled  triangle,  the  side 
opposite  the  right  angle  is  called  the  Hy- 
potenuse ;    and    the  other  two  sides    are 
called  the  Legs,  and  sometimes  the  Base 
and  Perpendicular. 

56.  When  an  angle  is  denoted  by  three 
letters,  of  which  one  stands  at  the  angular 
point,  and  the  other  two- on  the  two  sides, 
that  which  stands  at  the  angular  point  is 
read  in  the  middle.     Thus,  DAE. 

57.  The  circumference  of  every  circle  is  supposed  to  be 
divided  into  360  equal  parts,  called  degrees ;  and  each  de- 
gree into  60  minutes,  each  minute  into  60  seconds,  and  so 
on.    Hence  a  semicircle  contains  180  degrees,  and  a  quad- 
rant 90  degrees. 

58.  The  measure  of  an  angle,  is  an  arc 

of  any  circle  contained  between  the  two  vvx^~*sVr 
lines  which  form  that*  angle,  the  angular  ;'  Nv/  \ 
point  being  the  centre ;  and  it  is  estimated  \  / 

by  the  number  of  degrees   contained  in         *'•« -•'' 

that  arc. 

59.  Lines,  or  chords,  are  said  to  be 
Equidistant  from  the  centre  of  a  circle, 
when  perpendiculars  drawn  to  them  from 
the  centre  are  equal. 

60.  And  the  right  line  on  which  the 
Greater  Perpendicular  falls,  is  said  to  be 
farther  from  the  centre. 

61.  An  Angle   in   a  segment   is  that 
which  is  contained  by  two  lines,  drawn 
from  any  point  in,  the  arc  of  the  segment, 
to  the  two  extremities  of  that  arc. 

6'2.  An  Angle  on  a  segment,  or  an  arc,  is  that  which  is 
contained  by  two  lines,  drawn  from  any  point  in  the  oppo 
sue  or  supplemental  part  of  the  circumference,  to  the  extre 
unties  of  the  arc,  and  containing  the  arc  between  them. 

6 
f 


32 


GEOMETRY. 


63.  An  Angle  at  the  circumference,  is 
that   whose  angular  point  or   summit   is 
anywhere  in  the  circumference.      And  an 
angle  at  the  centre,  is  that  whose  angular 
point  is  at  the  centre. 

64.  A  right-lined  figure  is  Inscrihed  in  a 
circle,  or  the  circle  Circumscribes  it,  when 
all  the  angular  points  of  the  figure  are  in 
the  circumference  of  the  circle. 

65.  A  right-lined  figure  Circumscribes  a 
circle,  or  the  circle  is  Inscribed  in  it,  when 
all  the  sides  of  the  figure  touch  the  circum- 
ference of  the  circle. 

66.  One  right-lined  figure  is  Inscribed  in 
another,  or  the  latter  Circumscribes  the 
former,  when  all  the  angular  points  of  the 
former  are  placed  in  the  sides  of  the  latter. 

67.  A  Secant  is  a  line  that  cuts  a  circle, 
lying  partly  within  and  partly  without  it. 


68.  Two  triangles,  or  other  right-lined  figures,  are  said 
to  be  mutually  equilateral,  when  all  the  sides  of  the  one  are 
equal  to  the  corresponding  sides  of  the  other,  each  to  each  : 
and  they  are  said  to  be  mutually  equiangular,  when  the 
angles  of  the  one  are  respectively  equal  to  those  of  the 
other. 

69.  Identical  figures,  are  such  as  are  both  mutually  equi- 
lateral and  equiangular ;  or  that  have  all  the  sides  and  all 
the  angles  of  the  one,  respectively  equal  to  all  the  sides  and 
all  the  angles  of  the  other,  each  to  each  ;  so  that,  if  the  one 
figure  were  applied  to,  or  laid  upon  the  other,  all  the  sides 
of  the  one  would  exactly  fall  upon  and  cover  all  the  sides 
of  the  other ;  the  two  becoming  as  it  were  but  one  and  the 
same  figure. 

70.  Similar  figures,  are  those  that  hnve  all  the  angles  of 
the  one  equal  to  all  the  angles  of  the  other,  each  to  each, 
and  the  sides  about  the  equal  angles  proportional. 

71.  The  Perimeter  of  a  figure,  is  the  sum  of  all  its  sides 
taken  together. 

72   A  Proposition,  is  something  which  is  either  proposed 


THEOKEMS. 


63 


to  be  done,  or  to  be  demonstrated,  and  is  either  a  problem 
or  a  theorem. 

73.  A  Problem,  is  something  proposed  to  be  done. 

71.  ATheorem,  is  something  proposed  to  be  demonstrated. 

75.  A  Lemma,   is  something  which  is  premised,  or  de- 
monstrated, in  order  to  render  what  follows  more  easy. 

76.  A  Corollary,   is  a  consequent  truth,  gained  imme- 
diately from  some  preceding  truth,  or  demonstration. 

77.  A  Scholium,  is  a  remark  or  observation  made  upon 
something  going  before  it. 


THEOREMS. 

1.  In  the  two  triangles  ABC,  DEF,  if 
the  side  AC  be  equal  to  the  side  DF,  and 
the  side  BC  equal  to  the  side  EF,  and  the 
angle  C  equal  to  the  angle  F ;  then  will 
the  two  triangles  be  identical,  or  equal  in 
all  respects. 

2.  Let  the  two  triangles  ABC,  DEF,  have  the  angle  A 
equal  to  the  angle  D,  the  angle  B  equal  to  the  angle  E,  and 
the  side  AB  equal  to  the  side  DE ;  then  these  two  triangle* 
will  be  identical. 

3.  If  the  triangle  ABC  have  the  side 
AC  equal  to  the  side  BC  ;  then  will  the 
angle  B  be  equal  to  the  angle  A. 

The  line  which  bisects  the  vertical  angle 
of  an  isosceles  triangle,  bisects  the  base, 
and  is  also  perpendicular  to  it. 

Every  equilateral  triangle,  is  also  equiangular,  or  has  all 
its  angles  equal. 

4.  If  the  triangle  ABC,  have  the  angle 
A  equal  to  the  angle  B,  it  will  also  have 
the  side  AC  equal  to  the  side  BC. 

Every  equiangular  triangle  is  also  equi- 
lateral. * 

5.  Let  the  two  triangles  ABC,  ABD, 
have  their  three  sides  respectively  equal, 
viz.  the  side  AB  equal  to  AB,  AC  to  AD, 
and  BC  to  BD  ;  then  shall  the  two  triangles 
be  identical,  or  have  their  angles  equal, 


GEOMETRY. 


viz.  those  angles  that  are  opposite  to  the  equal  sides ;  viz 
the  angle  BAG  to  the  angle  BAD,  the  angle  ABC  to  the 
angle  ABD,  and  the  angle  C  to  the  angle  D. 

6.  Let  the  line  AB  meet  the  line  CD  ; 
then  will  the  two  angles  ABC,  ABD,  taken 
together,  be  equal  to  two  right  angles.  — 


7.  Let  the  two  lines  AB,  CD,  intersect 
in  the  point  E  ;  then  will  the  angle  A  EC 
be  equal  to  the  angle  BED,  and  the  angle 
AED  equal  to  the  angle  CEB. 

8.  Let  ABC  be  a  triangle,  having  the 
side  AB  produced  to  D  ;  then  will  the  out- 
ward angle  CBD  be  greater  than  either 
of  the  inward  opposite  angles  A  or  C. 


9.  Let  ABC  be  a  triangle,  having  the 
side  AB  greater  than  the  side  AC ;  then 
will  the  angle  ACB,  opposite  the  greater 
side  AB,  be  greater  than  the  angle  B,  op- 
posite the  less  side  AC. 

10.  Let  ABC  be  a  triangle ;  then  will 
the  sum  of  any  two  of  its  sides  be  greater 
than  the  third  side ;  as,  for  instance,  AC 
-f  CB  greater  than  AB. 


1 1 .  Let  ABC  be  a  triangle ;  then  will  the 
difference  of  any  two  sides,  as  AB — AC, 
be  less  than  the  third  side  BC. 


12.  Let  the  line  EF  cut  the  two  parallel 
lines  AB,  CD ;  then  will  the  angle  AEF 
be  equal  to  the  alternate  angle  EFD. 

13.  Let  the  line  EF,  cutting  the  two 
lines  AB,  CD,  make  the  alternate  angles 
AEF,  DFE,  equal  to  each  other ;  then 
wii!  AB  be  parallel  to  CD. 


65 


14.  Let  the  line  EF  cut  the  two  paral- 
lel lines  AB,  CD  ;  then  will  the  outward 
angle  EGB  be  equal  to  the  inward  opposite 
an  trie  GHD,  on  the  same  side  of  the  line 
KF;   and  the  two  inward  angles  BGH, 
GHD,  taken  together,  will  be  equal  to  two 
right  angles. 

15.  Let  the  lines  AB,  CD,  be  each  of 
them  parallel  to  the  line  EF  ;  then  shall 
the   lines  AB,  CD,   be   parallel  to  each 
other. 


16.  Let  the  side  AB,  of  the  triangle 
A.BC,  be  produced  to  D ;  then  will  the  out- 
ward angle  CBD  be  equal  to  the  sum  of 
ihe  two  inward  opposite  angles  A  and  C. 

17.  Let  ABC   be   any  plane  triangle; 
then  the  sumof  the  three  angles  A-fB  +  C 
is  equal  to  two  right  angles. 

If  two  angles  in  one  triangle,  be  equal 
to  two  angles  in  another  triangle,  the  third 
angles  will  also  be  equal,  and  the  two  triangles  equiangii 
lar. 

If  one  angle  in  one  triangle,  be  equal  to  one  angle  in 
another,  the  sums  of  the  remaining  angles  will  also  be 
equal. 

If  one  angle  of  a  triangle  be  right,  the  sum  of  the  other 
two  will  also  be  equal  to  a  right  angle,  and  each  of  them 
singly  will  be  acute,  or  less  than  a  right  angle. 

The  two  least  angles  of  every  triangle  are  acute,  or  eacQ 
less  than  a  right  angle. 

18.  Let  ABC  D  be  a  quadrangle ;    then 
the  sum  of  the  four  inward  angles,  A-f-B 
-fC-f  D  is  equal  to  four  right  angles. 


19.  Let  ABCDE  be  any  figure;  then 
the  sum  of  all  its  inward  angles,  A  +  B-f- 
C  +  D  +  E,  is  equal  to  twice  as  many 
right  angles,  wanting  four,  as  the  figure 
has  sides. 

6* 


66 


GEOMETKY. 


20.  Let  A.,  B,  C,  &c.,  be  the  outward 
angles  of  any  polygon,  made  by.  producing 
all  the  sides;  then  will  the  sum   A  +  B  + 
C-j-D  +  E,  of  all  those  outward  angles,  be 
equal  to  four  right  angles. 

21.  I%AB,  AC,  AD,'  <fec.,  be  lines  drawn 
from  the  given  point  A,  to  the  indefinite 

ine  BE,  of  which  AB  is  perpendicular; 
then  shall  the  perpendicular  AB  be  less 
than  AC,  and  AC  less  than  AD,  &c. 

22.  Let  ABCD   be  a  parallelogram,  of 
which  the  diagonal  is  BD ;    then  will  its 
opposite  sides  and  angles  be  equal  to  each 
other,  and  the  diagonal  BD  will  divide  it 
into  two  equal  parts,  or  triangles. 

If  one  angle  of  a  parallelogram  be  a  right  angle,  all  the 
other  three  will  also  be  right  angles,  and  the  parallelogram 
a  rectangle. 

The  sum  of  any  two  adjacent  angles  of  a  parallelogram 
is  equal  to  two  right  angles. 

2,3.  Let  ABCD  be  a  quadrangle,  having  the  opposite  sides 
equal,  namely,  the  side  AB  equal  to  DC,  and  AD  equal  to 
BC  ;  then  shall  these  equal  sides  be  also  parallel,  and  the 
figure  a  parallelogram. 

24.  Let  AB,  DC,  be  two  equal  and  parallel  lines;    then 
will  the  lines  AD,  BC,  which  join  their  extremes,  be  also 
equal  and  parallel. 

25.  Let  ABCD,  ABEF,  be  two  paral- 
lelograms, and  ABC,  ABF,  two  triangles, 
standing  on  the  same  base  AB,  and  between 
ihe  same  parallels  AB,  DE ;  then  will  the 
parallelogram  ABCD  be  equal  to  the  paral- 
lelogram ABEF,  and  the   triangle  ABC 
equal  to  the  triangle  ABF. 

Parallelograms,  or  triangles,  having  the  same  base  and 
altitude,  are  equal.  For  the  altitude  is  the  same  as  the 
perpendicular  or  distance  between  the  two  parallels,  which 
is  everywhere  equal,  by  the  definition  of  parallels. 

Parallelograms,  or  triangles,  having  equal  bases  and  :il- 
titudes,  are  equal.  For  if  the  one  figure  be  applied  wilh 
its  base  on  the  other,  the  bases  will  coincide  or  be  the  same, 
because  they  are  equal:  and  so  the  two  figures,  having  the 
same  base  and  altitude,  are  equal. 


n    f     H    G 

DO 


THKOUKMS. 

26.  Let  ABCD  be  a  parallelogram,  and 
ABE  a  triangle,  on  the  same  base  AH,  and 
between  the  same  parallels  AB,  I)l<- ;  then 
will  the  parallelogram  ABCI)   be  double 
the  triangle  ABE,  or  the  triangle  half  the 
parallelogram. 

A  triangle  is  equal  to  half  a  parallelogram  of  the  same 
base  and  altitude,  because  the  altitude  is  the  perpendicular 
distance  between  the  parallels,  which  is  everywhere  equal, 
by  the  definition  of  parallels. 

If  the  base  of  a  parallelogram  be  half  that  of  a  triangle, 
of  the  same  altitude,  or  the  base  of  a  triangle  be  double  that 
of  the  parallelogram,  the  two  figures  will  be  equal  to  each 
other. 

27.  Let  BD,   FH,  be   two  rectangles, 
having  the  sides  AB,  BC,  equal   to   the 
sides  EF,  FG,  each  to  each;  then  will  the 
rectangle  BD  be   equal  to  the  rectangle 
FH. 

28.  Let  AC   be  a  parallelogram,  BD  a 
diagonal,  EIF  parallel  to  AB  or  DC,  and 
OIH  parallel  to  AD  or  BC,  making  AI, 
1C,   complements   to   the    parallelograms 
EG,  HF,  which  are  about  the  diagonal  DB: 

then  will  the  complement  AI  be  equal  to  the  complement 
1C. 

29.  Let  AD  be  the  one  line,  and  AB 
the  other,  divided  into  the  parts  AE,  EF, 
FB ;  then  will  the  rectangle  contained  by 
AD  and  AB,  be  equal  to  the  sum  of  the 
rectangles  of  AD  and  AE,  and  AD  and 

EF,  and  AD  and  FB :  thus  expressed,  AD  .  AB=AD  .  AE 
+AD  .  EF+AD  .  FB.* 

If  a  right  line  be  divided  into  any  two  parts,  the  square 
of  the  whole  line,  is  equal  to  both  the  rectangles  of  the 
whole  line  and  each  of  the  parts. 

*  Instead  of  the  mark  X  »  a  point  is  often  used ;  thus,  length  X 
breadth  =  area,  is  the  same  as  length  .  breadth  =  area.  Instead  of  Jhe 
parenthesis,  a  stroke  is  often  used;  thus,  (first -j-  last)  -7-  2=  arith- 
metical mean,  is  the  same  thing  as  first  -f-  last  -f-  2  •=  arithmetical 
mean.  For  the  square  root  this  mark  ^/  is  sometimes  used,  and  for  the 
cube  root  /,  &c. 


68 


GEOMETRY. 


30.  Let  the  line  AB  be  the  sum  of  any 

two  lines  AC,  CB  ;    then  will  the  square  A 

of  AB  be  equal  to  the  squares  of  AC,  CB, 

together  with  twice  the  rectangle  of  AC 

.  CB.     That  is,  ABa=ACa+CB3-f  2AO  B 

.CB. 

If  a  line  be  divided  into  two  equal  parts  ;  the  square  of 
the  whole  line  will  be  equal  to  four  times  the  square  of  half 
the  line. 

31.  Let  AC,  BC,  be  any  two  lines,  and 
AB  their  difference ;  then  will  the  square 
of  AB  be  less  than  the  squares  of  AC,  BC, 
by  twice  the  rectangle  of  AC  and  BC. 
Or,  AB3  =  AC3  -f  BC2  —  2AC  .  BC. 

32.  Let  AB,  AC,  be  any  two  unequal 
lines;     then    will    the   difference    of  the 
squares  of  AB,  AC,  be  equal  to  a  rect- 
angle   under   their   sum    and    difference. 
That  is,  AB3— -  AC3  =  AB  -f-  AC  . 
AB  —  AC. 

33.  Let   ABC    be    a   right-angled   tri- 
angle, having  the  right  angle  at  C  ;  then 
will  the  square  of  the  hypotenuse  AB,  be 
equal  to  the  sum  pf  the  squares  of  the 
other  two  sides  AC,  CB.     Or  AB3=AC3 
-fBC3. 

34.  Let  ABC  be  any  triangle,  having  CD 
perpendicular  to  AB  ;  then  will  the  differ- 
ence of  the  squares  of  AC,  BC,  be  equal 
to  the  difference  of  the   squares  of  AD, 

BD  ;  that  is,  AC2— BC3=AD2— BD3.          A  u  B  D  A  B 

35.  Let  ABC  be  a  triangle,  obtuse-angled  at  B,  and  CD 
perpendicular  to  AB.;  ihen  will  the  square  of  AC  be  greater 
than  the  squares  of  AB,  BC,  by  twice  the  rectangle  of  AB, 
BD.     That  is,  AC3=AB2+BC2-f  2AB  .  BD. 

36.  Let  ABC  be  a  triangle,  having  the  angle  A  acute,  and 
CD  perpendicular  to  AB  ;  then  will  the  square  of  BC,  ba 
less  than  the  squares  of  AB,  AC,  by-twice  the  rectangle  of 
AB,  AD.     That  is,  BC2=AB3+AC3— 2AD    AB. 


THEOREMS. 


69 


37.  Let  ABC  be  a  triangle,  and  CD  the 
line  drawn  from  the  vertex  to  the  middle 
of  the  base  AB,  bisecting  it  into  the  two 
equal  parts  AD,  DB  :  then  will  the  sum 
of  the  squares  of  AC,  CB,  be  equal  to  twice 
the  sum  of  the  square  of  CD,  AD  ;  or  ACa 
-|-(;B8=2CD2+2ADa. 

38.  Let  ABC  be  an  isosceles  triangle, 
and  CD  a  line  drawn  from  the  vertex  to 
any  point  D  in  the  base :  then  will  the 
square  of  AC,  be  equal  to  the  square  of 
CD,  together  with  the  rectangle  of  AD  and 

DB.  That  is  ACS=CD2+AD  .  DB. 

39.  Let  ABCD    be    a    parallelogram, 
whose  diagonals  intersect  each  other  in  E  : 
then  will  AE  be  equal  to  EC,  and  BE  to 
ED ;  and  the  sum  of  the  squares  of  AC, 
BD,  will  be  equal  to  the  sum  of  the  squares 
of  AB,  BC,  CD,  DA.     That  is, 

AE=EC,  and  BE=ED, 
and  AC2-f  BD*=AB2+BC2+CDa+DA2 

40.  Let  AB  be  any  chord  in  a  circle, 
and  CD  a  line  drawn  from  the  centre  C  to 
the  chord.    Then,  if  the  chord  be  bisected 
in  the  point  D,  CD  will  be  perpendicular 
to  AB. 

41.  Let  ABC  be  a  circle,  and  D  a  point 
within  it ;  then  if  any  three  lines,  DA,  DB, 

DC,  drawn  from  the  point  D  to  the  cir- 
cumference, be  equal  to  each  other,  the 
point  D  will  be  the  centre. 

42.  Let  two  circles  touch  one  another  internally  in  the 
point ;  then  will  the  point  and  the  centres  of  those  circles 
be  all  in  the  same  right  line. 

43.  Let  two  circles  touch  one  another  externally  at  the 
point ;  then  will  the  point  of  contact  and  the  centres  of  the 
two  circles  be  all  in  the  same  right  line. 

44.  Let  AB,  CD,  be  any  two  chords  at 
equal  distances  from  the  centre  G ;  then 
will  these  two  chords  AB,  CD,  be  equal 
to  each  other. 


70 


GEOMETRY. 


45.  Let  the  line  ADB  be  perpendicular 
to  the  radius  CB  of  a  circle ;  then  shall 
AB  touch  the  circle  in  the  point  D  only. 

46.  Let  AB  be  a  tangent  to  a  circle, 
and  CD  a  chord  drawn  from  the  point  of 
contact  C  ;  then  is  the  angle  BCD  mea- 
sured by  half  the  arc  CFD,  and  the  angle 
ACD  measured  by  half  the  arc  CGD. 

47.  Let  BAG  be  an  angle  at  the  circum- 
ference ;  it  has  for  its  measure,  half  the 
arc  BC  which  subtends  it. 

48.  Let  C  and  D  be  two  angles  in  the 
same  segment  ACDB,  or,   which  is  the 
same  thing,  standing  on  the  supplemental 
arc  AEB  ;  then  will  the  angle  C  be  equal 
to  the  angle  D. 

49.  Let  C  be  an  angle  at  the  centre  C, 
and  D  an  angle  at  the  circumference,  both 
standing  on  the  same  arc  or  same  chord 
AB  ;  then  will  the  angle  C  be  double  of 
the  angle  D,  or  the  angle  D  equal  to  half 
the  angle  C. 

50.  If  ABC  or  ADC  be  a  semicircle, 
then  any  angle  D  in  that  semicircle,  is  a 
right  angle. 

51.  If  AB  be  a  tangent,  and  AC  a  chord, 
and  D  any  angle  in  the  alternate  segment 
ADC  ;  then  will  the  angle  D  be  equal  to 
the  angle  BAG  made  by  the  tangent  and 
chord  of  the  arc  AEG. 

52.  Let  ABCD  be  any  quadrilateral  in- 
scribed in  a  circle  ;  then  shall  the  sum  of 
the  two  opposite  angles  A  and  C,  or  B 
and  D,  be  equal  to  two  right  angles. 


THEOREMS. 


71 


53.  If  the  side  AB,  of  the  quadrilateral 
ABCD,  inscribed  in  a  circle,  be  produced 
to  E  :  the  outward  angle  DBE  will  be 
equal  to  the  inward  opposite  angle  C. 


54.  Let   the   two  chords  AB,  CD  be 
parallel  ;  then  will  the  arcs  AC,  BD,  be 
equal;  or  AC=BD. 

55.  Let  the  tangent  ABC  be  parallel  to 
the  chord  DF ;  then  are  the  arcs  BD,  BF, 
equal;  that  is,  BD=BF. 

56.  Let  the  two  chords  AB,  CD,  inter- 
sect at  the  point  E ;  then  the  angle  AEC, 
or  DEB,  is  measured  by  half  the  sum  of 
the  two  arcs  AC,  DB. 

57.  Let  the  angle  E  be  formed  by  two 
secants  EAB  and  ECD ;    this    angle    is 
measured  by  half  the  difference  of  the  two 
arcs  AC,  DB,  intercepted  by  the  two  se- 
cants. 

58.  Let  EB,  ED,  be  two  tangents  to  a 
circle  at  the  points  A,  C  ;  then  the  angle 
E  is  measured  by  half  the  difference  of  the 
two  arcs  CFA,  CGA. 


59.  Let  the  two  lines  AB,  CD,  meet 
each  other  in  E ;  then  the  rectangle  of  AE, 
EB,  will  be  equal  to  the  rectangle  of  CE, 
ED.  Or,  AE  .  EB  =  CE  .  ED. 

When  one  of  the  lines  in  the  se- 
cond case,  as  DE,  by  revolving  about 
the  point  E,  comes  into  the  position 
of  the  tangent  EC  or  ED,  the  two 
points  C  and  D  running  into  one ; 


72 


GEOMETRY. 


then  the  rectangle  of  CE,  ED,  becomes  the  square  of  CF 
because  CE  and  DE  are  then  equal.     Consequently,  th* 
rectangle  of  the  parts  of  the  secant,  AE  .  EB,  is  equal  to 
the  square  of  the  tangent,  CE3. 

60.  Let  CD  be  the  perpendicular,  and 
CE  the  diameter  of  the  circle  about  the 
triangle  ABC ;    then   the   rectangle   CA, 
CB  is  =  the  rectangle  CD  .  CE. 

61.  Let  CD  bisect  the  angle  C  of  the 
triangle  ABC  ;  then  the  square  CD2-f  the 
rectangle   AD  .  DB    is  =  the    rectangle 
AC  .  CB. 

62.  Let  ABCD  be  any  quadrilateral  in- 
scribed in  a  circle,  and  AC,  BD,  its  two 
diagonals  ;  then  the  rectangle  AC  .  BD  is 
=  the  rectangle  AB  .  DC  +  the  rectangle 
AD  .  BC. 

63.  Let  the  two  triangles  ADC,  DEF, 
have  the  same  altitude,  or  be  between  the 
same  parallels  AE,  CF ;  then  is  the  sur- 
face of  the  triangle  ADC,  to  the  surface 
of  the  triangle  DEF,  as  the  base  AD  is 
to  the  base  DE.    Or,  AD  :  DE  :  :  the  tri- 
angle ADC  :  the  triangle  DEF. 

64.  Let  ABC,  BEF,  be  two  triangles 
having  the  equal  bases  AB,  BE,  and  whose 
altitudes  are  the  perpendiculars  CG,  FH  ; 
then  will  the  triang-ie  ABC  :  the  triangle 
BEF  : :  CG  :  FH. 

Triangles  and  parallelograms,  when  their  bases  are  equal, 
are  to  each  other  as  their  altitudes ;  and  by  the  foregoing 
one,  when  their  altitudes  are  equal,  they  are  to  each  other 
as  their  bases  ;  therefore,  universally,  when  neither  are 
equal,  they  are  to  each  other  in  the  compound  ratio,  or  as 
the  rectangle  or  product  of  their  bases  and  altitudes. 

•  65.  Let  the  four  lines  A,  B,  C,  D,  be 
proportionals,  or  A  :  B  : :  C  :  D ; « then 
will  the  rectangle  of  A  and  D  be  equal  to 
the  rectangle  of  B  and  C  ;  or  the  rectangle 
i  .  D  -  B  .  C. 


THEOUEMS. 


73 


66.  Let  DE  he  parallel  to  the  side  BC 
of  the  trianolc  ABC  ;   then  will  AD  :  DB 
: :  AE  :  EC. 

AB  :  AC  : :  AD  :  AE, 
AB  :  AC  : :  BD  :  CE. 

67.  Let  the  angle  ACB,  of  the  triangle 
\.BC,  be  bisected  by  the  line  CD,  making 
.he  angle  r  equal  to  the  angle  s  :  then  will 
the  segment  AD  be  to  the  segment  DB, 
as  the  side  AC  is  to  the  side  CB.    Or,  AD 
:  DB  :  :  AC  :  CB. 

68.  In  the  two  triangles  ABC,  DEF,  if 
AB  :  DE  : :  AC  :  DF  : :  BC  :  EF ;  the 

two  triangles  will  have  their  correspond- 
ing angles  equal. 

69.  Let  ABC,  DEF,  be  two  triangles, 
having  the. angle  A  =  the  angle  D,  and 
the  sides  AB,  AC,  proportional  to  the  sides 
DE,  DF ;  then  will  the  triangle  ABC  be 
equiangular  with  the  triangle  DEF. 

70.  Let  ABC  be  a  right-angled  tri- 
angle, and  CD  a  perpendicular  from 
the  right  angle  C  to  the  hypotenuse 
AB  ;   then  will 

CD  be  a  mean  proportional  between  AD  and  DB ; 
AC  a  mean  proportional  between  AB  and  AD  ; 
BC  a  mean  proportional  between  AB  and  BD. 

71.  All  similar  figures  are  to  each  other,  as  the  squares 
of  their  like  sides. 

72.  Similar  figures  inscribed  in  circles,  have  their  like 
sides,  and  also  their  whole  perimeters,  in  the  same  ratio 
as  the  diameters  of  the  circles  in  which  they  arc  inscribed. 

73.  Similar  figures  inscribed  in  circles  are  to  each  other 
as  the  squares  of  the  diameters  of  those  circles. 

74.  The  circumferences  of  all  circles  are  to  each  other 
as  their  diameters. 

75.  The  areas  or  spaces  of  circles,  are  to  each  other  as 
the  squares  of  (heir  diameters;  or  of  their  radii. 

7(1.    The  area  of  any  circle,  is  equal  to  the  rectangle  of 
Half  its  circumference  and  half  its  diameter. 


1    To  bisect  a  line  AB  ;  that  is,  to  divide  it  into  two 
equal  parts. 

c 

From  the  two  centres  A  and  B,  with 
any  equal  radii,  describe  arcs  of  circles, 
intersecting  each  other  in  C  and  D  ; 
and  draw  the  line  CD,  which  will  bi- 
sect the  given  line  AB  in  the  point  E. 

2.    To  bisect  an  angle  BAG. 

From  the  centre  A,  with  any  radius, 
describe  an  arc  cutting  off  the  equal 
lines  AD,  AE ;  and  from  the  two  centres 
D,  E,  with  the  same  radius,  describe 
arcs  intersecting  in  F  ;  then  draw  AF, 
which  will  bisect  the  angle  A  as  re- 
quired. 

3.  At  a  given  point  C,  in  a  line  AB,  to  erect  a  per 
pendicular. 

From  the  given  point  C,  with  any  ra-  ^ 

dius,  cut  off  any  equal  parts  CD,  CE,  of 
the  given  line ;  and,  from  the  two  centres 
D  and  E,  with  any  one  radius,  describe 
arcs  intersecting  in  F ;  then  join  CF, 
tvhich  will  be  perpendicular  as  required. 

OTHERWISE. 

When  the  given  point  C  is  near  the  end  of  the  line. 

From  any  point  D,  assumed  above 
the  line,  as  a  centre,  through  the  given 
point  C  describe  a  circle,  cutting  the 
given  line  at  E  ;  and  through  E  and 
the  centre  D,  draw  the  diameter  EDF; 
then  join  CF,  which  will  be  the  per- 
oendicular  required. 


A  E 


PROBLEMS. 


75 


4.  From  the  giver,  point  A;  to  let  fall  a  perpendicular  on 
a  given  line  BC. 

From  the  given  point  A  as  a  centre, 
w'uh  any  convenient  radius,  describe  an 
arc,  cutting  the  given  line  at  the  two 
points  D  and  E  ;  and  from  the  two  cen- 
tres D,  E,  with  any  radius,  describe  two 
arcs,  intersecting  at  F ;  then  draw  AGF, 
which  will  be  perpendicular  to  BC  as  re- 
quired. 

OTHERWISE. 

When  the  given  point  is  nearly  opposite  the  end  of  the 
line. 

From  any  point  D,  in  the  given  line  . 

BC,  as  a  centre,  describe  the  arc  of  a  cir- 
cle through  the  given  point  A,  cutting 
BC  in  E  :  and  from  the  centre  E,  with 
the  radius  EA,  describe  another  arc,  cut- 
ting the  former  in  F;  then  draw  AGF, 
which  will  be  perpendicular  to  BC  as  re- 
quired. 

5.  At  a  given  point  A,  in  a  line  AB,  to  make  an  angle 

equal  to  a  given  angle  C. 

E 

From  the  centres  A  and  C,  with  any 
one  radius,  describe  the  arcs  DE,  FG. 
Then,  with  radius  DE,  and  centre  F,  de- 
scribe an  arc,  cutting  FG  in  G.  Through 
G  draw  the  line  AG,  and  it  will  form  the 

angle  required. 

A  F    B 

6.  Through  a  given  point  A,  to  draw  a  line  parallel  to  a 

s:iven  line  BC. 


From  the  given  point  A  draw  a  line  AD 
to  any  point  in  the  given  line  BC.  Then 
draw  the  line  EAF,  making  the  angle  at  A 
equal  to  the  angle  at  D  (by  prob.  5) ;  so 
shall  EF  be  parallel  to  BC  as  required. 


E  A 


76 


GEOMETRY. 


D 


7.  To  divide  a  line  AB  into  any  proposed  number  of 
equal  parts. 

Draw  any  other  line  AC,  forming  any 
angle  with  the  given  line  AB ;  on  which 
set  off  as  many  of  any  equal  parts  AD, 
DE,  EG,   FC,   as  the  line  AB  is  to  be 
divided  into.    JoinBC;  parallel  to  which    — r 
draw  the  other  lines  FG,  EH,  DI ;  then 
.these  will  divide  AB  in  the  manner  required. — For  those 
parallel  lines  divide  both  the  sides  AB,  AC,  proportionally. 

8.  To  find  a  third  proportional  to  two  given  lines  AB,  AC 

Place  the  two  given  lines  AB,  AC,    A — B 

forming  any  angle  at  A  ;  and  in  AB  take 
also  AD  equal  to  AC.  Join  BC,  and 
draw  DE  parallel  to  it ;  so  will  AE  be 
the  third  proportional  sought. 

9.  To  find  a  fourth  proportional  to  three  lines,  AB,  AC, 

AD. 

Place  two   of  the  given   lines   AB,    A B 

AC,  making  any  angle  at  A;  also 
place  AD  on  AB.  Join  BC  ;  and  pa- 
rallel  to  it  draw  DE  ;  so  shall  AE  be 
the  fourth  proportional  as  required. 


D     B 


10.  To  find  a  mean  proportional  between  two  lines,  AB 

BC. 

Place  AB,  BC,  joined  in  one  straight' 
line  AC  :  on  which,  as  a  diameter,  de- 
scribe the  semicircle  ADC  ;  to  meet  which 
erect  the  perpendicular  BD  ;  and  it  will 
be  the  mean  proportional  sought,  between 
AB  and  BC. 


B C 


O     B     C 


11.  To  find  the,  centre,  of  a  circle. 


Draw  any  chord  AB ;  and  bisect  it 
perpendicularly  with  the  line  CD,  which 
will  be  a  diameter.  Therefore  CD  bi- 
sected in  O,  will  give  the  centre,  as  re- 
quirec 


PROBLEMS. 


77 


12.  To  describe  the  circumference  of  a  circle  thiough 
three  given  points,  A,  B,  C. 

From  the  middle  point  B  draw  chords 
BA,  BC,  to  the  two  other  points,  and  bi- 
sect these  chords  perpendicularly  by  lines 
meeting  in  O,  which  will  be  the  centre. 
Then  from  the  centre  O,  at  the  distance 
of  any  one  of  the  points,  as  OA,  describe 
a  circle,  and  it  will  pass  through  the  two 
other  points  B,  C,  as  required. 


13.  To  draw  a  tangent  to  a  circle,  through  a  given 
point  A. 

When  the  given  point  A  is  in  the  cir- 
cumference of  the  circle  :  join  A  and  the 
centre  0  ;  perpendicular  to  which  draw 
BAG,  and  it  will  be  the  tangent. 


14.  On  a  given  line  B  to  describe  a  segment  of  a  circle, 
to  contain  a  given  angle  C. 

At  the  ends  of  the  given  line  make 
angles  DAB,  DBA,  each  equal  to  the 
given  angle  C.  Then  draw  AE,  BE  per- 
pendicular to  AD,  BD ;  and  with  the 
centre  E,  and  radius  EA  or  EB,  describe 
a  circle  ;  so  shall  AFB  be  the  segment 
required,  as  any  angle  F  made  in  it  will 
be  equal  to  the  given  angle  C. 


15.  To  cut  off  a  segment  from  a  circle,  that  shall  contain 
a  given  angle  C. 

Draw  any  tangent  AB  to  the  given  cir- 
cle ;  and  a  chord  AD  to  make  the  angle 
DAB  equal  to  the  given  angle  C  ;  then 
DEA  will  be  the  segment  required,  any 
angle  E  made  in  it  being  equal  to  the 
given  angle  O 

7* 


76 


GEOMETRY. 


16.  To  make  a  triangle  with  three  given  lines,  AB,  AC 
BC. 


With  the  centre  A,  and  distance  AC, 
describe  an  arc.  With  the  centre  B,  and 
distance  BC,  describe  another  arc,  cutting 
the  former  in  C.  Draw  AB,  BC,  and 
ABC  will  be  the  triangle  required. 


17.  To  inscribe  a  circle  in  a  given  triangle  ABC. 


Bisect  any  two  angles  A  and  B,  with 
the  two  lines  AD,  BD.  From  the  inter- 
section D,  which  will  be  the  centre  of  the 
circle,  draw  the  perpendiculars  DB,  DF, 
DG,  and  they  will  be  the  radii  of  the  cir- 
cle required. 

18.  To  describe  a  circle  about  a  given 
triangle  ABC. 

Bisect  any  two  sides  with  two  per- 
pendiculars DE,  DF,  and  their  inter- 
section D  will  be  the  centre. 


19.   To  inscribe  an  equilateral  triangle  in  a  given  circle 

Through  the  centre  C  draw  any  di- 
ameter AB.  From  the  point  B  as  a 
centre,  with  the  radius  BC  of  the 
given  circle,  describe  an  arc  DCE. 
Join  AD,  AE,  DE,  and  ADE  is  the 
equilateral  triangle  sought. 

20.   To  inscribe  a  square  in  a  given  circle. 

Draw  two  diameters  AC,  BD,  cross- 
ing at  right  angles  in  the  centre  E. 
Then  join  the  four  extremities  A,  B, 
C,  D,  with  right  lines,  and  these  will 
form  the  inscribed  square  ABCD. 


79 


21.  To  describe  a  square  about  a  given  circle. 

Draw  two  diameters  AC,  BD,  cross- 
ing at  right  angles  in  the  centre  E. 
Then  through  their  four  extremities 
draw  FG,  III,  parallel  to  AC,  and  FI, 
GH,  parallel  to  BD,  and  they  will 
form  the  square  FGHI.  I  »  H 

22.  To  inscribe  a  circle  in  a  regular  polygon. 

A 

Bisect  any  two  sides  of  the  poly- 
gon by  the  perpendiculars  GO,  FO, 
and  their  intersection  O  will  be  the 
centre  of  the  inscribed  circle,  and  OG 
or  OF  will  be  the  radius. 


23.   To  describe  a  circle  about  a  regular  polygon. 

Bisect  any  two  of  the  angles,  C 
and  D,  with  the  lines  CO,  DO  ;  then 
their  intersection  O  will  be  the  centre 
of  the  circumscribing  circle  ;  and  OC, 

or  OD,  will  be  the  radius. 

• 

24.   To  make  a  square  equal  to  the  sum  of  two  or  mart 

given  squares. 

Let  AB  and  AC  be  the  sides  of  two 
given  squares.  Draw  two  indefinite 
lines  AP,  AQ,  at  right  angles  to  each 
other ;  in  which  place  the  sides  AB, 
AC,  of  the  given  squares  ;  join  BC  ; 
then  a  square  described  on  BC  will  be 
equal  to  the  sum  of  the  two  squares 
described  on  AB  and  AC. 


P    B 


25.   To  make  a  square  equal  to  the  difference  of  two  givtn 

squares. 

Let  AT^  and  AC,  taken  in  the  same 
straight  L;ie,  be  equal  to  the  sides  of 
the  two  given  squares.  From  the 
centre  A,  with  the  distance  AB,  de- 
Bcribe  a  circle;  ami  make  CD  perpen- 
dicular to  AB,  meeting  the  circumference  in  D;  so  shall  a 


c  B 


sc 


GEOMETRY. 


square  described  on -CD  be  equal  to  AD2 — AC2,  or  AB9—- 
AC2,  as  required. 

26.    To  make  a  triangle  equal  to  a  given  pentagon 
ABCDE. 

D 

Draw  DA  and  DB,  and  also  EF, 
CG,  parallel  to  them,  meeting  AB  pro- 
duced at  F  and  G  :  then  draw  DF  and 
DG  ;  so  shall  the  triangle  DFG  be 
equal  to  the  given  pentagon  ABCDE. 

F         A        B      G 

27.   To  make  a  square  equal  to  a  given  rectangle  ABCD 

Produce  one  side  AB,  till  BE  be 
equal  to  the  other  side  BC.  On  AE 
as  a  diameter  describe  a  circle,  meet- 
ing BC  produced  at  F ;  then  will  BF 
be  the  side  of  a  square  BFGH,  equal 
to  the  given  rectangle  BD,  as  required. 


r> 


A    H 


APPENDIX  TO  GEOMETRY. 


INSTRUMENTS. 

28.  To  facilitate  the  construction  of  geometrical  figures 
we  add    a  short  description  of  a  few  useful  instruments 
which  do  not  belong  to  the  common  pocket-case. 

29.  Let  there  be  a  flat  ruler,  AB,  from       R 
one  to  two  feet  in  length,  for  which  the 
comnjon   Gunter's   scale  may   be  substi- 
tuted ;   and,  secondly,  a  triangular  piece 

of  wood,  «,  6,  c,  flat,  and  about  the  same 

thickness  as  the  ruler :  the  sides  ab  and 

V  of  which  are  equal  to  one  another,  and  form  a  right 

angle  at  b.    For  the  convenience  of  sliding,  there  is  usually 

a  hole  in  the  middle  of  the  triangle,  as  may  be  seen  in  the 

figure. 

30.  By  me#ns  of  these  simple  instruments  many  very 
useful  geometrical  problems  may  be  performed.     Thus,  to 
draw  a  line  through  a  given  point  parallel  to  a  given  line. 


INSTRUMENTS.  51 

Lay  the  triangle  on  the*  paper  so  that  one  of  its  sides  will 
coincide  with  the  given  line  to  which  the  parallel  is  to  be 
drawn  ;  then,  keeping  the  triangle  steady,  lay  the  ruler  on 
the  paper,  with  its  edge  applied  to  either  of  the  other  sides 
of  the  triangle ;  then;  keeping  the  ruler  firm,  move  the  tri- 
angle along  its  edge,  up  or  down,  to  the  given  point ;  the 
side  of  the  triangle  which  was  placed  on  the  given  line  will 
always  keep  parallel  to  itself,  and  hence  a  parallel  may  be 
drawn  through  the  given  point. 

31.  To  erect  a  perpendicular  on  a  given  line,  and  from 
any  given  point  in  that  line,  we  have  only  to  apply  the 
ruler  to  the  given  line,  and  place  the  triangle  so,  that  its 
right  angle  shall  touch  the  given  point  in  the  line,  and  one 
of  the  sides  about  the  right  angle,  placed  to  the  edge  of  the 
ruler — the  other  side  will  give  the  perpendicular  required. 

32.  If  the  given  point  be  either  above  or  below  the  line, 
the  process  is  equally  easy.    Place  one  of  the  sides  of  the 
triangle  about  the  right  angle  on  the  given  line,  and  the 
ruler  on  the  side  opposite  the  right  angle,  then  slide  the  tri- 
angle on  the  edge  of  the  ruler  till  the  given  point  from 
which  the  perpendicular  is  to  be  drawn  is  on  the  other  side, 
then  this  side  will  give  the  perpendicular. 

33.  Other  problems  may  be  performed  with  these  instru- 
ments, the  method  of  doing  which  it  will  be  easy  for  the 
reader  to  contrive  for  himself. 

34.  When  arcs  of  circles  of  great  diameter  are  to  be 
drawn,  the  use  of  a  compass  may  be  substituted  by  .a  very 
simple  contrivance.     Draw  the  chord  of  the  arc  to  be  de- 
scribed, and  place  a  pin  at  each  ex- 
tremity,  A   and    B,   then   place   two 

rulers  jointed  at  C,  and  forming  an 
angle,  ACB  =  the  supplement  of  half 
the  given  number  of  degrees  ;  that  is 
to  say,  the  number  of  degrees  which  the  arc  whose  chord 
given  is  to  contain,  is  to  be  halved,  and  this  half  being  sub- 
tracted from  180  degrees,  will  give  the  degrees  whicli  form 
the  angle  at  which  the  rulers  are  placed,  that  is;  the  angle 
ACB.  This  being  done,  the  edges  of  the  rulers  are  moved 
along  against  the  pins,  and  a  pencil  at  C  will  describe  the 
arc  required. 

35.  Large  circles  may  be   described  by  a  contrivance 
equally  simple.     On  an  axle,  a  foot  or  a  foot  and  a  half 


82  GEOMETRY. 

long,  there  are  placed  two  wheels,  M  and 
F,  of  which  one  is  fixed  to  the  axle,  name- 
ly, M,  and  the  other  is  capable  of  being 
shifted  to  different  parts  of  the  axle,  and, 
by  means  of  a  thumb-screw,  made  capable 
of  being  fixed  at  any  point  on  the  axle. 
These  wheels  are  of  different  diameters,  say  of  3  and  6 
inches,  the  fixed  wheel  F  being  the  largest.  This  instru- 
ment being  moved  on  the  paper,  the  circles  M  and  F  will 
roll,  and  describe  circles  of  different  radi  .  the  axle  will  al- 
ways point  to  the  centre  of  these  circle*,  and  there  will  be 
this  proportion : 

As  the  diameter  of  the  larg-e  wheel 

Is  to  the  difference  of  the  diameters  of  the  two  wheels, 

So  is  the  radius  of  the  circle  to  be  described  by  the 

large  wheel 
To  the  distance  of  the  two  wheels  on  the  axle. 

36.  If  the  diameters  of  the  wheels  are  as  above  stated, 
and  it  is  required  to  describe  a  circle  of  3  feet  radius,  then 
from  the  above  proportion  we  have  6:6  —  3  :  :  3  feet  or 
36  inches  :  18  inches  =  the  distance  of  the  two  wheels,  to 
describe  a  circle  6  feel  in  diameter. 

37.  It  may  be  observed,  that  it  will  be  best  to  make  the 
difference  of  the  wheels  greater  if  large  circles  are  to  be  de- 
scribed, as  then  a  shorter  instrument  will  serve  the  purpose. 

38.  We  will  conclude  this  appendix,  by  making  a  few 
remarks  on  the  Diagonal  Scale  and  Sector,  the  great  use 
of  the  lattei  jf  which,  especially,  is  seldom  explained  to 
the  young  mechanic. 

39.  The  diagonal  scale  to  be  found  on  the  plain  scale  in 
common  pocket-cases  of  instruments,  is  a  contrivance  for 
measuring  very  small  divisions  of  lines ;   as,  for  instance, 
hundredth  parts  of  an  inch. 

40.  Suppose  the  accompanying  cut  to       „     E       A 
represent   an   enlarged  view   of  two  di- 
visions   of  the    diagonal   scale,    and   the 

bottom  and  top  lines  to  be  divided  into 
two  parts,  each  representing  the  tenth 
part  of  an  inch.  Now,  the  perpendicular 
lines  BC,  AD,  are  each  divided  into  ten 
equal  parts,  which  are  joined  by  the 
crossing  lines,  1,  2,  3,  4,  &c.,  and  the  diagonals  BF,  DE, 


± 


INSTRUMENTS.  83 

nre  drawn  as  in  the  figure.  Now,  as  the  division  FC  is 
the  tenth  part  of  an  inch,  and  as  the  line  FB  continually 
approaches  nearer  and  nearer  to  BC,  till  it  meets  it  in  B, 
it  will  follow,  that  the  part  of  the  line  1  cut  off  by  this 
diagonal  will  be  a  tenth  part  of  FC,  because  Bl  is  only  one- 
tenth  part  of  BC ;  so,  likewise,  2  will  represent  two-tenth 
parts,  3,  three-tenth  parts,  and  so  on  to  9,  which  is  nine- 
tenth  parts,  and  10,  ten-tenth  parts,  or  the  whole  tenth  of 
an  inch ;  so  that,  by  means  of  this  diagonal,  we  arrive  at 
divisions  equal  to  tenth  parts  of  tenth  parts  of  an  inch,  or 
hundredths  of  an  inch.  With  this  consideration,  an  exami- 
nation of  the  scale  itself  will  easily  show  the  whole  matter. 
It  may  be  observed,  that  if  half  an  inch  and  the  quarter  of 
an  inch  be  divided,  in  the  same  manner,  into  tenths  and 
tenths  of  tenths,  we  may  get  thus  two-hundredth  and  four- 
hundredth  parts  of  an  inch. 


THE    SECTOR. 

41.  THIS  very  useful  instrument  consists  of  two  equal 
rulers   each  six  inches    long,  joined  together  by  a  brass 
folding  joint.    These  rulers  are  generally  made  of  boxwood 
or  ivory;  and  on  the  face  of  the  instrument,  several  lines 
or  scales  are  engraven.     Some  of  these  linjes  or  scales  pro- 
ceed from  the  centre  of  the  joint,  and  are  called  sectorial 
lines,  to  distinguish  them  from  others  which  are  drawn 
parallel  to  the  edge  of  the  instrument,  similar  to  those  on 
the  common  Gunter's  scale. 

42.  The  sectorial  lines  are  drawn  twice  on  the  same  face 
of  the  instrument;  that  is  to  say,  each  line  is  drawn  on  both 
legs.     Those  on  each  face  are, 

A  scale  of  equal  parts,  marked  L, 
A  line  of  chords,  marked  C, 

A  line  of  secants,  marked  S, 

A  line  of  polygons,  marked       P,  or  Pol. 
These  sectnrial  lines  are  marked  on  one  face  of  the  instru- 
ment; and  on  the  other  there  are  the  following: 
A  line  of  sines,  marked  S, 

A  line  of  tangents,  marked        T, 
A  line  of  tangents  to  a  less  radius,  marked  L 


84  GEOMETRY. 

Tins  last  line  is  intended  to  supply  the  defect  of  the  former 
and  extends  from  about  45  to  75  degrees. 

43.  The  lines  of  chords,  sines,  tangents,  and  secants,  but 
not  the  line  of  polygons,  are  numbered  from  the  centre, 
and  are  so  disposed  as  to  form  equal  angles  at  the  centre ; 
and  it  follows  from  this,  that  at  whatever  distance  the  sec- 
tor is  opened,  the  angles  which  the  lines  form,  will  always 
be  respectively  equal.    The  distance,  therefore,  between  10 
and  10,  on  the  two  lines  marked  L,  will  be  equal  to  the 
distance  of  60  and  60  on  the  two  lines  of  chords,  and  also 
to  90  and  90  on  the  two  lines  of  sines,  &c.,  at  any  particu- 
lar opening  of  the  sector. 

44.  Any  extent  measured  with  a  pair  of  compasses,  from 
the  centre  of  the  joint  to  any  division  on  the  sectorial  lines, 
is  called  a  lateral  distance;    and  any  extent  taken  from  a 
point  in  a  line  on  the  one  leg,  to  the  like  point  on  the 
similar  line  on   the   other  leg,  is   called  a  transverse  or 
parallel  distance. 

With  these  remarks,  we  shall  now  proceed  to  explain  the 
use  of  the  sector,  in  so  far  as  it  is  likely  to  be  serviceable  to 
mechanics. 

USE    OF    THE    LINE    OF    LINES. 

45.  This  line,  as  was  before  observed,  is  marked  L,  and 
its  uses  are, 

To  Divide  a  line  into  any  number  of  equal  parts  :  Take 
the  length  of  th^line  by  the  compasses,  and  placing  one  of 
the  points  on  that  number  in  the  line  of  lines  which  denotes 
the  number  of  parts  into  which  the  given  line  is  to  be 
divided,  open  the  sector  till  the  other  point  of  the  com- 
passes touches  the  same  division  on  the  line  of  lines  marked 
on  the  other  leg ;  then,  the  sector  being  kept  at  the  same 
width,  the  distance  from  1  on  the  line  L  on  the  one  leg, 
to  1  on  the  line  L  on  the  other,  will  give  the  length  of  one 
of  the  equal  divisions  of  the  given  line  to  oe  divided. 
Thus,  to  divide  a  given  line  into  seven  equal  parts  : — take 
the  length  of  the  given  line  with  the  compasses,  and  setting 
one  point  on  7,  on  the  line  L  of  one  of  the  legs,  move  the 
other  leg  out  until  the  other  point  of  the  compasses  touch 
7  on  the  line  I,  of  that  leg ;  this  may  be  called  the  trans- 
verse distance  of  7  on  the  line  of  lines.  Now,  keeping  the 
sector  at  the  same  opening,  the  transverse  distance  of  1 
will  be  the  length  of  one  of  the  7  equal  divisions  of  the 


INSTRUMENTS.  f>5 

giver  line ;  the  transverse  distance  of  2  will  be  two  of  these 
divisions,  &c. 

46.  It  will  sometimes  happen,  that  the  line  to  be  divided 
will  be  too  long  for  the  largest  opening  of  the  sector ;  and 
in  this  case  we  take  the  half,  or  third,  or  fourth  of  the  line, 
as  the  case  may  be  ;  then  the  transverse  distance  of  1  to  1, 
will  be  a  half,  a  third,  or  a  fourth  of  the  required  equal  part. 

47.  To  divide  a  given  line  into  any  number  of  parts  that 
shall  have  a  certain  relation  or  proportion  to  each  other: 
Take  the  length  of  the  whole  line  to  be  divided,  and  placing 
one  point  of  the  compasses  at  that  division   on  the  line 
of  lines   on   one  leg  of  the  instrument  which  expresses 
the;  smn  of  all  the  parts  into  which  the  given  line  is  to  be 
divided,  and   open  the  sector  till  the  other  point   of  the 
compasses  is  on  the  corresponding  division  on  the  line  of 
lines  of  the  other  leg.     This  is  evidently  making  the  sum 
of  the  parts  into  which  the  given  line  is  to  be  divided  a 
transverse  distance;  and  when  this  is  done,  the  proportional 
parts  will  be  found  by  taking,  with  the  same  opening  of 
the  sector,  the  transverse  distances  of  the  parts  required.-— 
To  divide  a  given  line  into  three  parts,  in  the  proportion 
of  2,  3,  4:  The  sum  of  these  is  9;   make  the  given  line  a 
transverse  distance  between  9  and  9  on  the  two  lines  of 
lines;  then  the  transverse  distances  of  the  several  numbers 
2,  3,  4,  will  give  the  proportional  parts  required. 

48.  To  find  a  fourth  proportional  to  three  given  lines: 
Take  the  lateral  distance  of  the  second,  and  make  it  the 
transverse  distance  of  the  first,  then  will  the  transverse  dis- 
tance of  the  third  be  the  lateral  distance  of  the  fourth;  then, 
let  there  be  given  6:3::  8, — make  the  lateral  distance  of 
3  the  transverse  distance  of  6 ;    then  will  the  transverse 
distance  of  8  be  the  lateral  distance  of  4,  the  fourth  propoi- 
tional  required. 

49.  This  sector  will  be  found  highly  serviceable  in  draw- 
ing plans.     For  instance,  if  it  is  wished  to  reduce  the  draw- 
ing of  a  steam  engine  from  a  scale  of  1^  inches  to  the  foot, 
to  another  of  f  to  the  foot.    Now,  in  lg  inches  there  are  */ 
parts;  so  that  the  drawing  will  be  reduced  in  the  propor- 
tion of  12  10  5.     Take  the  lateral  distance  of  5,  and  keep 
the  compasses  at  this  opening ;  then  open  the  sector  till  the 
points  of  the  compasses  mark  the  transverse  distance  of  12 ; 
keep  now  the  sector  at  this  opening,  and  any  measure  taken 
on  the  drawing,  to  be  copied  and  laid  off  on  the  sector  as  a 

8 


86  MECHANICAL    DRAWING 

lateral  distance, — the  transverse  distance  taken  from  that 
point  will  give  the  corresponding  measure  to  be  laid  down 
in  the  new  drawing. 

50.  If  the  length  of  the  side  of  a  triangle,  of  which  we 
have  the  drawing,  is   to  he   reckoned   45 ;  what  are  the 
lengths  of  the  other  two  sides  ?     Take  the  length  of  the 
side  given,  by  the  compasses,  and  open  the  sector  till  the 
measure  be  the  transverse  distance  of  45  to  45 ;    then  the 
lengths  of  the  other  sides  being  applied  transversely,  will 
give  their  numerical  lengths. 

USE    OF    THE    LINE    OF    CHORDS. 

51.  By  means  of  the  sector,  we  may  dispense  with  the 
protractor.     Thus,  to  lay  down  an  angle  of  any  number  of 
degrees: — take  the  radius  of  the  circle  on  the  compasses, 
and  open  the  sector  till  this  becomes  the  transverse  distance 
of  60  on  the  line  of  chords ;   then  take  the  transverse  dis- 
tance of  the  required  number  of  degrees,  keeping  the  sec- 
tor at  the  same  opening;  and  this  transverse  distance  being 
marked  off  on  an  arc  of  the  circle  whose  radius  was  taken, 
will  be  the  required  number  of  degrees. 

We  will  not  enter  farther  on  the  use  of  the  sectorial  lines, 
as  what  we  have  said  will,  we  hope,  be  found  sufficient  for 
the  purposes  of  the  practical  mechanic. 


MECHANICAL  DRAWING  AND  PERSPECTIVE. 

52.  A  FLAT  rectangular  board  is  first  to  be  provided,  of 
any 'convenient  size,  as  from  18  to  30  inches  long,  and  from 
16  to  24  inches  broad.     It  may  be  made  of  fir,  plane  tree, 
or  mahogany ;  its  face  must  be  planed  smooth  and  flat,  and 
the  sides  and  ends  as  nearly  as  possible  at  right  angles  to 
each  other — the  bottom  of  the  board  and  the  left  side  should 
be  made  perfectly  so  ;  and  this  corner  shouL!  be  marked, 
so  that  the  stock  of  the  square  may  be  always  applied  to 
the  bottom   and  left  hand  side  of  the  board.     To  prevent 
the  board  from  casting,  it  is  usual  to  pannel  it  on  the  back 
or  on  the  sides. 

53.  A  T  square  must  also  be  provided,  such  that  by 


AND    PERSPECTIVE.  87 

• 

means  of  a  thumb-screw  fixed  in  the  stock,  it  may  be  made 
to  answer  either  the  purposes  of  a  common  square,  or  bevel, 
—the  one-half  of  the  stock  being  movable  about  the  screw, 
and  the  other  fixed  at  right  angles  on  the  blade.  The  blade 
ought  to  be  somewhat  flexible,  and  equal  in  length  to  the 
length  of  the  board. 

54.  Besides  these,  there  will  be  required  a  case  of  mathe- 
matical instruments  ;  in  the  selection  of  which  it  should  be 
observed,  that  the  bow  compass  is  more  frequently  defect- 
ive than  any  of  the  other  instruments.     After  using  any  of 
the  ink  feet,  they  should  be  dried  ;  and  if  they  do  not  draw 
properly,  they  ought  to  be  sharpened  and  brought  to  an 
equal  length  in  the  blade,  by  grinding  on  a  hone. 

55.  The  colours  most  useful  are,  Indian  ink,  gamboge, 
Prussian  blue,  vermilion,  and  lake.    With  these,  all  colours 
necessary  for   drawing   machinery  or   buildings    may  be 
made ;  so  that,  instead  of  purchasing  a  box  of  colours,  we 
would  advise  that  those  for  whom  this  book  is  intended 
should  procure  these  cakes  separately  :  the*  gamboge  may 
be  bought  from  an  apothecary — a  pennyworth  will  serve 
a  lifetime.      In  choosing  the  rest,  they  should  be  rubbed 
against  the  teeth,  and  those  which  feel  smoothest  are  of  the 
best  quality. 

56.  Hair  pencils  will  also  be  necessary,  made  of  camel's 
hair,  and  of  various  sizes.     They  ought  to  taper  gradually 
to  a  point  when  wet  in  the  mouth,  and  should,  after  being 
pressed  against  the  finger,  spring  back. 

57.  Black-lead   pencils  will  also  be  necessary.     They 
ought  not  to  be  very  soft,  nor  so  hard  that  their  traces  can- 
not be  easily  erased  by  the  Indian  rubber      In  choosing 
paper,  that  which  will  best  suit  this  kind  of  drawing  is 
thick,  and  has  a  hardish  feel,  not  very  smooth  on  the  sur- 
face, yet  free  from  knots. 

58.  The  paper  on  which  the  drawing  is  to  be  made,  must 
be  chosen  of  a  good  quality  and  convenient  size.    It  is  then 
to  be  wet  with  a  sponge  and  clean  water,  on  the  opposite 
side  from  that  on  which  the  drawing  is  to  lie  made.     When 
the  paper  absorbs  the  water,  which  may  be  seen  by  the 
wetted  side  becoming  dim,  as  its  surface  is  viewed  slantwise 
against  the  light,    it  is  to  be  laid  on  the  drawing  board 
v/ith  the  wetted  side  next  the  board.     About  half  an  inch 
must  be  turned  up  ou  a  straight  edge  all  round  the  paper 


88  MECHANICAL    DRAWING 

and  then  fastened  on  the  board.  This  is  done  because  th« 
paper  when  wet  is  enlarged,  and  the  edges  being  fixed  or. 
the  board,  act  as  stretchers  when  the  paper  contracts  by 
drying.  To  prevent  the  paper  from  contracting  before  the 
paste  has  been  sufficiently  fastened  by  drying,  the  paper  is 
usually  wet  on  the  upper  surface,  to  within  half  an  inch  of 
the  paste  mark.  When  the  paper  is  thoroughly  dried,  it 
will  be  found  to  lie  firmly  and  equally  on  the  board,  and  is 
then  fit  for  use. 

59.  If  the  drawing  is  to  be  made  from  a  copy,  we  ought 
first  to  consider  what  scale  it  is  to  be  drawn  to.     If  it  is  to 
be  equal  in  size  to,  or  larger  than  the  copy ;  and  a  scale 
should  be  made  accordingly,  by  which  the  dimensions  of 
the  several  parts  of  the  drawing  are  to  be  regulated.     The 
diagonal  scale,  a  simple  and  beautiful  contrivance,  will  be 
here  found  of  great  use  for  the  more  minute  divisions  ;  and 
whenever  the  drawing  is  to  be  made  to  a  scale  of  1.  inch, 
£  inch,  |  inch  to  the  foot,  a  scale  should  be  drawn  of  20  or 
30  equal  parts  ;  the  last  of  which  should  be  subdirided  into 
12,  and  a  diagonal  scale  formed  on  the  same  principles  as 
the  common  one,  but  with  eight  parallels  and  12  diagonals, 
to  express  inches  and  eighths  of  an  inch.    For  making  such 
scales  to  any  proportion,  the  line  L  on  the  sector  will  be 
found  very  convenient. 

60.  Great  care  should  be  taken  in  the  penciling,  that  an 
accurate  outline  be  drawn,  for  on  this  much  of  the  value 
of  the  picture  will  depend.     The  pencil  marks  should  be 
distinct,  yet  not  heavy,  and  the  use  of  the  rubber  should  be 
avoided  as  much   as  possible,  as  its  frequent  application 
ruffles   the   surface   of  the   paper.     The  methods  already 
given  for  constructing   geometrical   figures    will   be   here 
found    applicable,  and  the  use  of  the  T  square,  parallel 
ruler,  &c.,  will  suggest  themselves  whenever  they  require 
to  be  employed. 

61.  The  drawing  thus  made  of  any  machine  or  building 
is  called  a  plan.     Plans  are  of  three  kinds — a  ground  plan, 
or  bird's-eye  view,  an  elevation  or  front  view,  and  a  per- 
spective plan. 

62.  When  a  view  is  taken  of  the  teeth  of  a  wheel,  with 
the  circumference  towards  the  eye,  the  teeth  appear  to  be 
nearer  as  they  are  removed  from  the  middle  point  of  the 
circumference  opposite  the  eye,  and  it  may  not  be  out  of 


AND    PERSPECTIVE.  ,  89 

place  here  to  give  the  method  of  representing  them  on 
paper: — IfABbe  the  circumference  of 
a  wheel  as  viewed  by  the  eye,  and  it  is  ^T 
required  to  represent  the  teeth  as  they 
appear  on  it.  Only  half  of  the  circum- 
ference can  be  seen  in  this  way  at  one 
time,  consequently  we  can  only  represent  the  half  of  the 
teeth.  On  AB  describe  a  semicircle,  which  divide  into 
half  as  many  equal  parts  as  the  wheel  has  teeth  ;  then  from 
each  of  these  points  of  division  draw  perpendiculars  to  the 
wheel  AB,  then  will  these  perpendiculars  mark  the  relative 
places  of  the  teeth. 

63.  When  the  outline  is  completed  in  pencil,  it  is  next 
to  be  carefully  gone  over  with  Indian  ink,  which  is  to  be 
rubbed  down  with  a  little  water,  on  a  plate  of  glass  01 
earthenware — so  as  to  be  sufficiently  fluid  to  flow  easily  out 
of  the  pen,  and  at  the  same,  time  have  a  sufficient  body  of 
colour.  While  drawing  the  ink  lines,  the  measurement 
should  all  be  repeated,  so  as  to  correct  any  error  that  may 
have  slipped  during  the  penciling.  The  screw  in  the 
drawing  pen  will  regulate  the  breadth  of  the  strokes  ;  which 
should  not  be  alike  heavy  ;  those  strokes  being  the  heaviest 
which  bound  the  dark  part  of  the  shades.  Should  any  line 
chance  to  be  wrong  drawn  with  the  ink,  it  may  be  taken 
out  by  means  of  a  sponge  and  water,  which  could  not  be 
done  if  common  writing  ink  were  employed. 

65.  In  preparing  for  colouring  it  is  to  be  observed,  that 
a  hair  pencil  is  to  be  fixed  at  each  end  of  a  small  piece  of 
wood,  made  in  the  form  of  a  common  pencil,  one  of  which 
is  to  be  used  with  colour,  and  the  other  with  water  only. 
If  the  colour  is  to  be  laid  on,  so  as  to  represent  a  flat  sur- 
face, it  ought  to  be  spread  on  equally,  and  there  is  here  no 
jise  for  the  water  brush  ;  but  if  it  is  to  represent  a  curved 
surface,  then  the  colour  is  to  be  laid  on  the  part  intended 
.o  be  shaded,   and   softened  towards  the  light  by  washing 
with  the  water  brush.     In  all  cases  it  should  be  borne  in 
*nind,  that  the  colour  ought  to  be  laid  on  very  thin,  other- 
wise it  will  be  more  difficult  to  manage,   and   will  never 
make  so  fine  a  drawing. 

66.  In  colours  even  of  the  best  quality,  we  sometimes 
meet  with  gritty  particles,  which  it  is  desirable  to  avoid 
Instead  of  rubbing  the  colour  on  a  plate  with  a  little  water 

8* 


90  MECHANICAL    DUAWING 

as  is  usual,  it  will  be  better  to  wet  the  colour,  and  rub  it  on 
the  point  of  the  forefinger,  letting  the  dissolved  part  drop 
off  the  finger  on  to  the  plate. 

67.  In  using  the  Indian  ink,  it  will  be  found  advanta- 
geous to  mix  it  with  a  little  blue  and  a  small  quantity  of  lake, 
which  renders  it  much  more  easily  wrought  with,  and  this 
is  the  more  desirable  as  it  is  the  most  frequently  used  of  all 
the  other  colours  in  Mechanical  Drawing,  the  shades  being 
all  made  with  this  colour. 

The  depth  and  extent  of  the  shades  will  depend  on 
various  circumstances — on  the  figure  of  the  .object  to  be 
shaded,  the  position  of  the  eye  of  the  observer,  and  the 
direction  in  which  the  light  comes,  <fce.  The  position  of 
the  eye  will  vary  the  proportionate  size  of  any  object  in  a 
picture  when  drawn  in  perspective.  Thus,  if  a  perspective 
view  of  a  steam  engine  is  given,  the  eye  being  supposed  to 
be  placed  opposite  the  end  nearest  the  nozzles,  an  inch  of 
the  nozzle  rod  will  appear  much  larger  than  an  inch  of  the 
pump  rod  which  feeds  the  cistern  ;  but  if  the  eye  is  sup- 
posed to  be  placed  opposite  the  other  end  of  the  engine, 
the  reverse  will  be  the  case.  But  in  drawing  elevations 
and  ground  plans  of  machinery,  every  part  of  the  machine 
is  drawn  to  the  proper  scale — an  inch  or  foot  in  one  part 
of  the  machine,  being  just  the  same  size  as  an  inch  or  foot 
in  any  other  part  of  the  machine.  So  that  by  measuring 
the  dimensions  of  any  part  of  the  drawing,  and  then  apply- 
ing the  compass  to  the  scale,  we  determine  the  real  size  of 
the  part  so  measured.  Whereas,  if  the  view  were  given  in 
perspective,  we  would  be  obliged  to  make  allowance  for  the 
effect  of  distance,  &c. 

68.  The  light  is  always  supposed  to  fall  on  the  picture 
at  an  angle  of  forty-five  degrees,  from  which  it  follows,  that 
the  shade  of  any  object,  which  is  intended  to- rise  from  the 
plane  of  the  picture,  or  appear  prominent,  will  just  be  equal 
in  length  to  the  prominence  of  the  object. 

69.  The  shades,  therefore,  should  be  as  exactly  measured 
as  any  other  part  of  the  drawing,  and  care  should  be  taken 
that  they  all  fall  in  the  proper  direction,  as  the  light  is  sup- 
posed to  come  from  one  point  only. 

70.  It  is  frequently  of  great  use  for  the  mechanic  to  take 
a  hasty  copy  of  a  drawing,  and  many  methods  have  been 
given  for  this  purpose — by  machines,   tracing,  &c.     Wa 
give  the  following  as  easy,  accurate,  and  convenient. 


AND    PEUSPECTIVK.  91 

Mix  equal  parts  of  turpentine  and  drying  oil,  and  with  a 
rag  lay  it  on  a  sheet  of  good  silk  paper,  allowing  the  paper 
to  lie  by  for  two  or  three  days  to  dry,  and  when  it  is  so  it  will 
be  fit  for  use.  To  use  it,  lay  it  on  the  drawing  to  be  copied, 
and  the  prepared  paper  being  nearly  transparent,  the  lines 
of  the  drawing  will  be  seen  through  it,  and  may  be  easily 
traced  with  a  black-lead  pencil.  The  lines  on  the  oiled 
paper  will  be  quite  distinct  when  it  is  laid  on  white  paper. 
Thus,  if  the  mechanic  has  little  time  to  spare,  he  may  take 
a  copy  and  lay  it  past  to  be  recopied  at  his  leisure. 

Care  and  perseverance  are  the  chief  requisites  for  attain- 
ing perfection  in  this  species  of  drawing.  Every  mechanic 
should  know  something  of  it,  so  that  he  may  the  better  un- 
derstand how  to  execute  plans  that  may  be  submitted  to 
him.  or  make  intelligible  to  others  any  invention  he  him- 
self may  make. 


CONIC   SECTIONS. 


DEFINITIONS. 

A  CONE  is  a  solid  figure  having  a  circle  for  its  base  and 
terminated  in  a  vertex  ;  it  may  be  conceived  to  be  formed 
by  the  revolution  of  a  triangle  about  one  of  its  sides. 

Conic  Sections  are  the  figures  made  by  a  plane  cutting  a 
cone.  According  to  the  different  positions  of  the  cutting 
plane  there  arise  five  different  figures  or  sections,  namely, 
a  triangle,  a  circle,  an  ellipse,  an  hyperbola,  and  a  parabola : 
the  last  three  of  which  only  are  peculiarly  called  Conic 


Sections.  If  the  cutting  plane  pass  through  the  vertex  of 
the  cone,  and  any  part  of  the  base,  the  section  will  be  a 
triangle  ;  as  VAB,  fig.  1.  If  the  plane  cut  the  cone  parallel 
*o  the  base,  or  make  no  angle  with  it,  the  section  will  be  a 
circle  ;  as  fig.  2.  The  section  DAB  is  an  ellipse  when  the 
cone  is  cut  obliquely  through  both  sides,  or  when  the  plane 
is  inclined  to  the  base  iu  a  less  angle  than  the  side  of  the 
cone  is,  fig.  3.  The  section  is  a  parabola,  when  the  cone 
is  cut  by  a  plane  parallel  to  the  side,  or  when  the  cutting 
plane  and  the  side  of  the  cone  make  equal  angles  with  the 
base,  fig.  4.  The  section  is  an  hyperbola,  when  the  cutting 
plane  makes  a  greater  angle  with  the  base  than  the  side  of 
the  cone  makes,  fig.  5.  And  if  all  the  sides  of  the  cone  be 
continued  through  the  vertex,  forming  an  opposite  equal 
cone,  and  the  plane  be  also  continued  to  cut  the  opposite 
cone,  this  latter  section  will  be  the  opposite  hyperbola  to 
ihe  former. 

92 


DEFINITIONS.  93 

The  Vertices  of  any  section,  are  the  points  where  the 
cutting  plane  meets  the  opposite  sides  of  the  cone,  or  the 
•ides  of  the  vertical  triangular  section. 

Hence  the  ellipse  and  the  opposite  hyperbolas,  have  each 
two  vertices  ;  but  the  parabola  only  one  ;  unless  we  consider 
the  other  as  at  an  infinite  distance. 

The  Axis,  or  Transverse  Diameter,  of  a  conic  section,  is 
the  line  or  distance  between  the  vertices. 

Hence  the  axis  of  a  parabola  is  infinite  in  length. 

The  centre  is  the  middle  of  the  axis. 

Hence  the  centre  of  a  parabola  is  infinitely  distant  from 
the  vertex.  And  of  an  ellipse,  the  axis  and  centre  lie 
within  the  curve ;  but  of  an  hyperbola,  the  axis  and  centre 
lie  without  it. 

A  Diameter  is  any  right  line  drawn  through  the  centre, 
and  terminated  on  each  side  by  the  curve  ;  and  the  extremi- 
ties of  the  diameter,  or  its  intersections  with  the  curve,  are 
its  vertices. 

Hence  all  the  diameters  of  a  parabola  are  parallel  to  the 
axis,  and  infinite  in  length.  Hence  also  every  diameter  of 
the  ellipse  and  hyperbola  has  two  vertices  ;  but  of  the  para- 
bola, only  one ;  unless  we  consider  the  other  as  at  an  infi- 
nite distance. 

The  Conjugate  to  any  diameter,  is  the  line  drawn  through 
the  centre,  and  parallel  to  the  tangent  of  the  curve  at  the 
vertex  of  the  diameter. 

Hence  the  conjugate  of  the  axis  is  perpendicular  to  it. 

An  Ordinate  to  any  diameter,  is  a  line  parallel  to  its  con- 
jugate, or  to  the  tangent  at  its  vertex,  and  terminated  by 
the  diameter  and  curve. 

Hence  the  ordinates  of  the  axis  are  perpendicular  to  it. 

An  Absciss  is  a  part  of  any  diameter  contained  between 
its  vertex  and  an  ordinate  to  it. 

Hence,  in  the  ellipse  and  hyperbola,  every  ordinate  has 
two  determinate  abscisses  ;  but  in  the  parabola  only  one  ; 
the  other  vertex  of  the  diameter  being  infinitely  distant. 

The  Parameter  of  any  diameter  is  a  third  proportional 
to  that  diameter  and  its  conjugate,  in  the  ellipse  and  hyper- 
bola, and  to  one  absciss  and  its  ordinate  in  the  parabola. 

The  Focus  is  the  point  in  the  axis  where  the  ordinate  is 
equal  to  half  the  parameter. 

The  ellipse  and  hyperbola  have  each  two  foci ;  but 
parabola  only  one. 


94  CONIC    SECTIONS. 

PROBLEMS  FOR  THE  CONIC  SECTIONS. 
THE  PARABOLA. 

1.  Given  two  abscisses  A  and  B,  together  with  the  ordi- 
nate  of  A,  to  find  the  ordinate  of  B. 

*/  absciss  B  x  ordinate  A  _ 

— — : T =  ordinate  B. 

\/  absciss  A 

Ex. — An  absciss  is  9,  and  its  ordinate  is  16,  it  is  required 
to  find  the  ordinate  of  another  absciss  36. 

^  36  X  16        6  X  16 

= =  32,  the  required  ordinate. 

V'    " 

2.  Given  the  ordinate  and  absciss,  required  the  para- 
meter. 

ordinate  3 

— - — : =  parameter. 

absciss 

Ex. — The  ordinate  being  12  and  absciss  6,  then, 

12 a        144 

— — = =  24  =  the  parameter  required. 

3.  To  find  the  length  of  the  curve  of  a  parabola,  cut  off 
by  a  double  ordinate  to  the  axis. 

v/  (ordin.8  -f  £  abs.2)  x  2  =  the  length  of  the  curve. 

Ex. — The  length  of  the  double  ordinate  being  12  and  the 
absciss  2,  then, 
\/  (6a  +  f  22)  x  2  =  12-858  =  the  length  of  the  curve. 

NOTE. — This  rule  is  sufficiently  correct  for  practice,  but 
will  not  apply  when  the  absciss  is  greater  than  the  half 
ordinate. 

THE  ELLIPSE. 

1.  To  find  an  ordinate,  we  have  the  proportion : 

As  the  transverse  axis  is  to  the  conjugate,  so  is  the  square 
root  of  the  product  of  the  two  abscisses,  to  the  ordinate. 

Ex. — The  transverse  axis  being  60,  the  conjugate  45,  the 
one  absciss  12,  and  the  other  48,  then, 

60  :  45  : :  </  (48  X  12)  :  18  =»  the  ordinate  required. 


PROBLEMS.  95 

2.  To  finci  the  absciss. 

v/  (the  half  conju. a  —  ordin. s)  x  trans,  axis 

conjugate  axis. 

distance  between  the  ordinate  and  centre  of  the  axis,  which 
being  added  to  the  half  axis,  will  give  the  greater  absciss 
or  being  subtracted,  will  give  the  shorter  absciss. 

Ex. — One  axis  being  20  and  the  other  15,  what  are  the 
abscisses  to  the  ordinate  whose  length  is  6. 

— — =  6  =  the  distance  from  the  centre, 

15 

wherefore  10  -f  6  =  16  *»  the  longer  absciss,  and  10 
—  6  =  4  =  the  shorter. 

3.  To  find  the  conjugate  axis. 

As  x/(one  absciss  x  other  absciss)  is  to  their  ordiaate, 
so  is  the  transverse  axis  to  the  conjugate. 

Ex. — The  transverse  axis  being  200,  the  ordinate  60 
one  absciss  is  40  and  the  other  160,  then, 

^/(leO  X  40)  :  60  : :  200  :  150  =  the  conjugate  axis. 

4.  To  find  the  transverse  axis. 

Take  the  square  root  of  the  difference  of  the  squares  of 
the  ordinate  and  half  conjugate,  and  add  to  this  the  half 
conjugate  if  the  lesser  absciss  is  used,  but  subtract  the  .half 
conjugate  if  the  greater  absciss  is  used.  In  either  case  call 
the  result  of  this  part  of  the  operation  M,  then, 

conjugate  x  absciss  x  M 

— -ji —  — =  transverse  axis, 

ordinate  * 

Ex. — If  the  ordinate  be  20,  the  lesser  absciss  14,  and  the 
conjugate  50,  then  by  the  above, 

s/(253  —  202)  -f  25  =  40  =  M. 

50  X  14  X  40 

— --^ —    — =  70  =  the  transverse  axis. 

5.  To  find  the  circumference  of  an  ellipse. 

sum  of  the  sq.  of  the  two  axes\ 

—  2—  —)  X  3-1416  =  circumfer 

Ex. — The  one  axis  being  24  and  the  other  18,  then. 

24s  -t-  183\ 

3. J  X  3-1416  =  66-643  =  circumference. 


86  CONIC    SECTIONS. 

THE  HYPERBOLA. 

1.  To  find  the  ordinate. 

As  the  transverse  axis  is  to  the  conjugate  ;  so  is  the  square 
root  of  the  product  of  the  two  abscisses,  to  the  ordinate. 

Ex. — The  transverse  axis  being  24,  the  conjugate  21,  and 
the  absciss  8  ;  then, 

24  :  21  : :  v/(32  X  8) :  14  =  the  ordinate. 

2.  To  find  the  abscisses. 

•v/ford.9  +  half  conjn.*)X  trans,  axis 

— =  distance  between 
conjugate 

the  ordin.  and  centre.  Then  this  distance,  added  to  the 
half  transverse,  gives  the  greater  absciss ;  or,  subtracted 
from  it,  the  less. 

Ex. — The  transverse  axis  being  40,  the  conjugate  32, 
and  the  ordinate  12;  then, 

x/(12a+  16')X40  .  . 

~— i =  25  =  distance  from  the  middle  of 

3Z 

the  transverse.  Wherefore,  25  -f  20  =  45  =  the  greatef 
absciss ;  and  25  —  20  =  5  =  the  lesser. 

3.  To  find  the  conjugate. 

ordinate  X  transverse  axis 

— =-: —    — " r  =  conjugate. 

v/(productof  the  abscisses) 

Ex. — The  transverse  axis  being  144,  the  lesser  absciss 
48,  and  its  ordinate  84 ;  then, 

84  x  144 
•v/(192  X  48)=          =         conJugate  required. 

4.  To  find  the  transverse. 

Take  the  half  conjugate,  and,  according  as  the  lesser  or 
greater  absciss  is  used,  add  it  to,  or  subtract  it  from,  the 
square  root  of  the  sum  of  the  squares  of  the  half  conjugal 
and  of  the  ordinate,  and  call  this  result  m ;   then, 

abscissa  x  conjugate  x  m 

P — *~f =  the  transverse  axis. 

ordinate2 

Ex. — The  conjugate  being  18,  the  lesser  absciss  10,  and 
'ts  ordinate  12;  then, 

9  +  */(9*  +  122)  =  9  -f  15  =  24  =  m; 

10  X  18  x  24 

— =  30  =  the  transverse  axis. 

Lm 


PROBLEMS. 


\ 


Descriptions  of  Conic  Sections  on  a  Plane. 

1.  Parabola.     Let  AB  be  a  v       „       w     . 
right  line  and  C  a  point  with- 
out it,  and  DKF  a  ruler  in  the 

same  plane  with  the  line  and 
point,  so  that  one  side,  as  DK, 
be  applied  to  AB,  and  KF 
coincide  with  the  point  C  ; 
on  F,  fix  one  end  of  the 
thread  FNC,  and  the  other  at 
the  point  C ;  and  Jet  part  of 
the  thread,  as  FN,  be  brought  to  the  side  KF  by  a  pin  N  , 
then  let  the  square  DKF,  be  removed  from  B  to  A,  applying 
its  side  DK  dose  to  BA;  and  in  the  mean  time  the  thread 
will  be  always  applied  to  the  side  KF;  and  by  the  motion 
of  the  pin  N  there  will  be  described  a  curve  called  a  semi- 
parabola.  Then  bringing  the  square  to  its  first  position 
moving  from  B  to  H  the  other  semi-parabola  will  be 
described. 

2.  Ellipse.     If  two  points,  as 
A  and  B,  be  taken  in  any  plane, 
and    in    them    is  fixed  a  thread 
longer  than  the  distance,  between 
them,  and  this  be   extended  by 
means  of  a  pin  C  ;  and  the  pin 

be  moved  round  from  any  point  till  it  return  back  again 
to  the  same  place,  the  thread  being  extended  all  the  while, 
the  figure  described  is  an  ellipse. 

3.  Hyperbola.     If  to  the  point  A,  one  end  of  the  ruler 
AB  be  placed,  so  that  about  that  point  as  a  centre  it  may 
freely  move ;    and  if  to  the  other  end  B  is  fixed  the  ex- 
tremity   of    the    thread 

BDC  shorter  than  the 
ruler  A13,  and  the  other 
end  of  the  thread  fixed 
in  the  point  C,  coincid- 
ing with  the  side  of  the 
ruler  AB  :n  the  same 
place  with  the  given 
point  A ;  let  part  of  the 
thread  BD  be  brought 
to  the  side  of  the  ruler 


98 


CONIC    SECTIONS. 


A.B  by  the  pin  D  ;  then  let  the  ruler  be  moved  about  the 
point  A  from  C  to  T,  the  thread  extended,  and  the  re- 
maining part  coinciding  with  the  side  of  the  ruler;  by  the 
motion  of  the  pin  D  a  semi-hyperbola  will  be  described 
The  ellipse  returns  into  itself:  but  the  parabola  and  hyper- 
bola are  unlimited. 


USEFUL  CURVES. 

* 

THE  Cycloid  is  a  very  useful  curve  ;  and  may  be  defined, 
the  curve  formed  by  a  nail  in  the  rim  of  a  wheel,  while  il 
moves  along  a  level  road.  The  cycloid  may  be  described 
on  paper,  thus : — If  the  circumference  of  a  circle  be  rolled 


on  a  right  line,  beginning  at  any  point  A,  and  continued 
till  the  same  point  A  arrive  at  the  line  again,  making  just 
one  revolution,  and  thereby  measuring  out  a  straight  line 
ABA  equal  to  the  circumference  of  the  circle,  while  the 
point  A  in  the  circumference  traces  out  a  curve  line 
ACAGA:  then  this  curve  is  called  a  cycloid;  and  some 
of  its  properties  are  contained  in  the  following  iemma: 

If  the  generating  or  revolving  circle  be  placed  in  the 
middle  of  the  cycloid,  its  diameter  coinciding  with  the  axis 
AB,  and  from  any  point  there  be  drawn  the  tangent  CF, 
the  ordinate  CDE  perpendicular  to  the  axis,  and  the  chord 
of  the  circle  AD  ;  then  the  chief  properties  are  these  : 

The  right  line         CD  =  the  circular  arc          AD  ; 

The  cycloidal  arc  AC    =  double  the  chord       AD  ; 

The  semi-cycloid  ACA=  double  the  diameter  AB,  and 

Tne  tangent  CF  is  parallel  to  the  chord  AD. 

If  the  ball  of  a  pendulum  be  made  to  move  in  a  cycloid, 
its  vibrations  will  be  isochronous,  or,  they  will  all  be  per- 
formed in  the  same  time.  The  cycloid  is  also  the  line  of 
swiftest  descent,  or,  a  body  will  fall  through  the  arc  of  this 
curve,  from  one  given  point  to  another,  in  less  time  than 
through  any  other  path.  See  Centre  of  Oscillation 


CONIC    SECTIONS.  99 

The  Catenary  is  that  curve  which  is  formed  by  a  chain 
or  chord  of  uniform  texture,  when  it  is  hung  upon  tw<? 
points,  and  left  to  hang  freely,  without  any  restraint.  It 
matters  not  whether  these  points  of  suspension  be  in  the 
same  horizontal  line  or  not,  or  whether  the  chain  be  slack 
or  tight;  still  the  fcurve  will  be  a  catenary.  A  knowledge 
of  this  curve  is  very  useful  in  the  construction  of  suspension 
bridges  See  the  chapter  on  Strength  of  Material*. 


MENSURATION. 


DEFINITIONS. 

To  the  definitions  in  geometry  the  lollowmg  are  aduud, 
in  order  to  make  the  subject  of  mensuration  understood. 

1.  A.  prism  is  a  solid,  of  which   the  sides  are  parallelo- 
grams,  and   the   ends   equal,    similar,   and    parallel    plane 
figures.     The  figure   of  the   ends   gives  the  name  to  the 
prism  ;  if  the  ends  are  triangular,  the  prism  is  triangular, 
&c.     If  the  sides  and  ends  of  a  prism  be  all  equal  squares, 
the  prism  is  called  a  cube  ;  and  if  the  base  or  ends  be  a 
parallelogram,  the  prism  is  called  a  parallelopipedon.    The 
cylinder  is  a  round  prism,  having  circular  ends. 

2.  The  pyramid  has  any  plane  figure  for  its  base,  and 
its  sides  triangles,  of  which  all  the  vertices  meet  in  a  point 
at  the  top,  called  the  vertex  of  the  pyramid. 

3.  A  sphere  or  globe  is  a  solid  bounded  by  one  continued 
surface,  every  point  of  which  surface  is  equally  distant  from 
a  point  within  the  sphere,  called  the  centre.    The  diameter 
or  axis  of  a  sphere,  is  any  line  which  passes  through  its 
centre,  and  is  terminated  at  both  ends  by  the  circumference 

4.  A  prismoid  has  its  two  ends  as  any  unlike  parallel 
plane   figures   of  the  same  number  of  sides  ;  the  upright 
sides  being  trapezoids. 

5.  A  spheroid  is  a  solid  resembling  the  figure  of  a  sphere, 
but  not  exactly  round — one  of  its  diameters  being  longer 
than  the  other ;  and,  likewise,  a  conoid  is  like  a  cone,  but 
has  not  its  sides  straight  lines  but  curved. 

6.  A  spindle  is  a  solid  formed  by  the  revolution  of  some 
curve  round  its  base. 

7.  The  axis  of  a  solid  is  a  straight  line  drawn  through 
the  solid,  from  the  middle  of  one  end  to  the  middle  of  the 
opposite  end. 

8.  The  height  of  a  solid  is  a  line  drawn  from  the  vertex, 
perpendicular  to  the  base,  or  the  plane  on  which  the  base  rests. 

9.  The  segment  of  a  solid  is  a  part  cut  off  by  a  plane, 
parallel  to  the  base  ;  and  the  frustum  is  the  part  remaining 
after  the  segment  is  cut  off. 

100 


MENSURATION. 


101 


SURFACES. 
1.  For  the  area  of  a  square,  rhombus,  or  rhomboid. 

Base  X  height  =  area. 

Ex. — The  base  of  a  rhombus  is  16,  the  height  9  ;  there- 
fore, 16  X  9  =  144  =  area. 

2.  For  the  area  of  a  triangle. 

5  (base  X  height)  =  area. 

Ex. — The  base  of  a  triangle  is  2j,  and  height  7<3  ;  there- 
fore, d  (2-25  X  7-5)  =  8-437,  the  area. 

3.  For  the  area  of  a  frapczoid. 
5  (sum  of  the  two  parallel  sides)  x  height  =  area. 
Ex. — In  a  trapezoid  one  of  the  parallel  sides  is  16s,  tb<? 
other  is  14],  and  the  height  or  perpendicular  distance  be- 
tween them  is  7  ;  therefore, 

I  (16-125  +  14-25)  x  7  =  106-3125,  the  area. 

4.  For  any  right-lined  figure  of  four  or  more  unequal 

sides. 

Divide  it  into  triangles,  by  lines  drawn  from  various 
angles ;  find  the  area  of  each  ;  then,  the  sum  of  these  areas 
will  be  the  area  of  the  whole  figure. 

5.  For  a  regular  polygon. 

Inscribe  a  circle  ;  then,  5  (radius  of  insc.  circle  x  length 
of  one  side  x  number  of  sides)  =  area. 

Ex. — In  a  polygon  of  8  sides,  the  length  of  a  side  is  16, 
and  radius  of  inscribed  circle  19-2  ;  then  £(3x16x8)  = 
1230,  the  area. 

The  following  table  will  greatly  facilitate  the  solution  of 
questions  connected  with  polygons. 


iid« 

Nime  of 

An?.  F  at 

Ang.  C  of 

Ana. 

A. 

B. 

c. 

3 

Trigon 

120° 

60° 

0-4330127 

2- 

1-73 

•579 

4 

Tetragon 

90 

90 

1-0000000 

1-41 

1-412 

•705 

6 

Pentagon 

72 

108 

1-7204774 

1-238  1-174 

•852 

6 

Hexagon 

60 

120 

25980762 

1-156 

=  ft»Jiu> 

=  Leo#* 

of  side. 

7 

Heptagon 

51  : 

128* 

3-6339124 

1-11 

•867 

1-16 

8 

Octagon 

45' 

135 

4-8284271 

1-08 

•765 

1-307 

9 

Nonagon 

40 

140 

6-1818242 

1-062-681 

1-47 

10 

Decagon 

36 

144 

7-6942088 

1-05 

•616 

1-625 

11 

Urulecagon 

32T8T 

147J\ 

9-3656405 

1-04 

•561 

1-777 

12 

Dodecagon 

30 

150 

11-1961524 

1-037 

•515625 

1-94 
1 

102  MENSURATION. 

The  first  column  of  this  table  gives  the  number  of  sides 
of  the  polygon ;  the  second,  the  name ;  the  uses  of  the 
third  and  fourth  will  be  explained  in  the  note  at  the  bottom 
of  the  page,*  and  the  uses  of  the  rest  will  appear  by  the  fol- 
lowing rules  and  examples.  The  answers  found  are  only 
approximate,  but  come  sufficiently  near  the  truth  for  all 
practical  purposes. 

Side  of  polygon  3  x  No.  column  AREA  =  area. 

Ex. — In  *a  figure  of  10  equal  sides,  the  length  of  one 
side  being  8,  we  have  8'2  =  8  x  8  =  64 ;  hence  64  X 
7-6942088  =  492-4293632  =  the  area. 

Take  the  length  of  a  perpendicular,  drawn  from  the 
centre  to  one  of  the  sides  of  a  polygon,  and  multiply  this 
by  the  numbers  in  column  A,  the  product  will  be  the  radius 
of  the  circle  that  contains  the  polygon. 

Ex. — If  the  length  of  a  perpendicular  drawn  from  the 
centre  to  one  of  the  sides  of  an  octagon  be  12,  then  12  x 
1*08  =  12-96  =  radius  of  circumscribing  circle. 

The  radius  of  a  circle  multiplied  by  the  number  in 
column  B,  will  give  the  length  of  the  side  of  the  correspond- 
ing polygon  which  that  circle  will  contain.  Suppose,  for 
an  octagon,  the  radius  of  a  circle  to  be  12-96,  then  12-96 
X  '765  =  9-9144  =  the  length  of  one  side  of  the  inscribed 
polygon  of  8  sides. 

The  length  of  the  side  of  a  polygon  multiplied  by  the 
corresponding  number  in  the  column  C,  will  give  the  radius 
of  circumscribing  circle.  Thus  the  length  of  one  side  of  a 
decagon  being  10;  then  10  x  1'625  =  16-25  =  radius 
of  circumscribing  circle. 

6.  For  the  circle. 
1st,  diameter  X  3-1416  =  circumference; 

*  The  third  and  fourth  columns  of  the  table  of  polygons  will  greatly 
facilitate  the  construction  of  these  figures  by  the  aid  of  the  sector.  Thus, 
if  it  be  required  to  describe  a  polygon  of  eight  sides,  then  look  in  column 
Angle  F,  opposite  Octagon,  and  you  find  45.  With  the  chord  of  60  on 
the  sector  as  radius  describe  a  circle,  then  taking  the  length  45  on  the 
same  line  of  the  sector,  mark  this  distance  off  on  the  circumference, 
which  being  repeated  round  the  circle,  will  give  the  points  of  junction 
of  the  sides  cf  the  octagon.  The  fourth  column  of  the  table  gives  the 
angle  in  degrees,  which  any  <  wo  adjoining  sides  of  the  respective  figurei 
make  with  each  other. 


MENSURATION. 


103 


,    circumference 
2</'  -TI416-     =diameler; 
3d,  £  circumference  x  radius  =  area. 
Ex. — In  a  circle  whose  diameter  is  14  inches,  we  have 
1st,  14  x  3-1416  =  43-9824,  the  circumference; 
43-9824 


*d, 


3-1416 


=  14,  the  diameter ; 
14 


8d,  diameter  •—•  2  =  radius,  so  ~  =  7  =  radius.    Then 
£  (43-9824)  X  7  =  153-9384,  the  area. 

7.  For  the  length  of  the  arc  of  a  circle. 

Radius  x  -079577  X  number  of  degrees  =  length  of  are. 
Ex. — If  the  radius  be  12,  ind  number  of  degrees  22,  then 
12  X  -079577  X  22  =  21-008328,  the  length. 

8.  For  the  area  of  a  circular  sector. 

Radius  X  5  length  of  arc. 

Ex. — The  radius  being  12,  and  length  of  arc  2 1-008328  ; 
then,  12  X  10-504164  =  126-049968,  the  area. 

9.  For  the  area  of  a  circular  segment. 

TABLE    OF    THE    AREAS    OF    CIRCULAR    SEGMENTS. 


H. 

AIM. 

H. 

Arta. 

H.              A 

H 

A-e». 

•01 

•001329 

•14 

•06683:5 

•27     -171089 

•40 

•293369 

•02 

•003748 

•15 

•073874 

•28     -100019 

•41 

•303187 

•03 

•006865 

•16 

•081112 

•29 

•189047 

•42 

•313041 

•04 

•010537 

•17 

•088535 

•30 

•198168 

•43 

•322928 

•05 

•014681 

•18 

•096134 

•31 

•207376 

•44 

•332843 

•06 

•019239 

•19 

•103900 

•32 

•216666 

•45 

•3-1  27*2 

•07 

•024168 

•20 

•111823 

•33 

•226033 

•46 

•352742 

•08 

•029435 

•21 

•119897 

•34 

•235  i73 

•47 

•362717 

•09 

•035011 

•2* 

•128113 

•35 

•244980 

•48 

•372704 

•10 

•040875 

•23 

•136465 

•36 

•251550 

•49 

•382099 

•11 

•047005 

•24 

.144944 

•37 

•264)78 

•50 

•392699 

•12 

•053385 

•25 

•153546 

•38 

•273861 

•001 

•000042 

•13 

•059999 

•26 

•162263 

•39 

•2S3592 

•002 

•000119 

This  may  be  done  easily  by  the  help  of  the  preceding 
table  ;  to  use  which,  divide  the  height  of  the  segment  by 
the.  diameter  of  the  circle,  and  look  for  the  quotient  in  the 


104  MENSURATION. 

column  H,  opposite  to  which  will  be  found  a  number  in 
column  AREA,  which  multiplied  by  the  square  of  the  dia- 
meter will  give  the  area  of  the  segment.  Should  the  height 
of  the  segment  be  greater  than  the  diameter,  find  by  the 
foregoing  rule  the  area  of  the  remaining  segment,  and  by 
subtracting  this  from  the  area  of  the  whole  circle,  the  area 
of  the  greater  segment  will  be  found. 

18 

EA — Let  the  height  be  18  and  diameter  48,  then — =  -37; 

which,  in  the  column  marked  H  in  col.  AREA,  corresponds 
to  -264178  ;  hence  482  x  '264178  =  608-6661  =  the  area. 

10.  For  the  area  of  a  cycloid. 
Area  of  generating  circle  x  3  =  area  of  cycloid. 
Ex. — The  diameter  of  generating  circle  being  10,  then 
£  (10  x  3-1416)  x  V  X  3  =  235-619,  the  area  of  cycloid. 

11.  For  the  area  of  a  parabola. 
(Base  x  height)  X  |  =  the  area. 
Ex. — The  base  being  20,  and  height  6  ;  then, 
20  x  6  X  |  =  80,  the  area. 

12.  'For  the  area  of  an  ellipse. 
(Long  axis  x  short  axis)  x  -7854  =  area. 
Ex. — The -greater  axis  being  300,  and  lesser  200  ;  then, 
300  x  200  x  '7854  =  47124,  the  area. 


§  SOLIDS. 
1.  For  the  surface  and  content  of  a  prism  or  cylinder. 

1st.  Area  of  two  ends  +  length  X  perimeter  =  surface. 

2d.  Area  of  base  x  height  =  content. 

The  circumference  of  a  cylinder  is  6,  and  its  length  9 
inches  ;  what  is  the  surface  and  content  ? 

The  area  of  each  end  is  2-85  ;  therefore  2  X  2'85  == 
5-7  =  the  area  of  the  two  ends,  and  then  5'7  -f-  (6  X  9) 
=  59-7  =  the  area  of  the  whole  cylinder.  Also,  2-85  X 
9  =  25-65  =  content. 

2.  For  a  cone  or  pyramid. 

1st.  k  (slant  height  x  perimeter  of  base)  +  area  of  base 
=  surface. 


MENSURATION. 


105 


2</.   3  (area  of  base  x  perpend,  height)  =  content. 

Ex. — Shun  height  is  10,  perimeter  of  base  16;  then,  £ 
(10  X  )6  =  80  -f  10  =  96,  surface  of  a  four-sided  pyra- 
mid, whose  side  at  the  base  is  4. 

The  area  of  the  base  of  a  cone  being  147-68,  and  per- 
pendicular height  14, 

Then  |  (U\  147*68)  =  689-17,  content. 

3.  For  a  cube  or  parallelopiped. 

Isf,  The  sum  of  tli£  areas  of  all  the  sides  =  surface. 

2d.    Length  X  breadth  x  depth  =  content. 

Ex — In  a  parallelopiped  the  length  30,  breadth  6,  and 
depth  4. 

30  x  6  x  4  =  720,  content,  and  648  =  the  surface. 

It  is  worthy  of  remembrance  that  one  cubic  foot  contains 
1728  cubic  inches.  22,000  cylindric,  3300  spherical  inches, 
and  66  conical.  The  cone,  sphere,  and  cylinder,  are  as  1, 
2,  and  3. 

4.  For  regular  or  platonic  bodies,  or  bodies  of  equal  sides 

1st.  Linear  edge  2x  tabular  number  of  figures  for  sur- 
face =  surface. 

2(1.  Linear  edge  3X  tabular  number  of  figures  for  soli- 
dity =  content. 


No.  of  Sides. 

Name. 

Multiplier  for 
Surface. 

"  Multiplier  for 
Solidity. 

4 
6 
8 
12 
20 

Tetrahedron, 
Hexahedron, 
Octahedron, 
Dodecahedron, 
Icosahedron, 

1-7320508 
6-0000000 
3-4641016 
20-6457288 
8-6602540 

0-1178513 
1-00000 
0-4714045 
7-6631189 
2-181695 

Ex. — In  an  Octahedron  the  length  of  the  ridge  of  a  side 
is  5,  therefore  5*  x  3-4641016  =  86-6025  =  surface,  and 
53  x  -4714045  =  58-9255,  the  solidity. 

5.  For  the  surface  of  a  sphere  and  segment. 

/Diameter  2X  3-1416  =  surface  of  the  whole  sphere. 

Ex. — If  the  diameter  be  36,  then  36"  x 3- 14 16=4071 '504 
square  inches  =  surface. 

Height  of  segment  X  diameter  of  sphere  x  3-1416  =* 
surface  of  segment. 


106  MENSURATION. 

Ex. — The  diameter  of  the  sphere   being  12,  and  the 
height  of  segment  6,  then 
6  Xl2  X  3-1416  =  226-1952  =  surface  of  spheric  segment. 

6.  For  the.  content  of  a  sphere  and  spheric  segment. 

Diameter  3X  0-5236  =  content. 

Ex. — If  the  diameter  of  a  sphere  be  2  inches,  then  28 
X  0-5236  =  4-1888  =  the  content. 

(radius  of  segment's  base  ax  3  -f-  height  of  segment8)  x 
height  x  -5236  =  content  of  segment. 

Ex. — If  the  height  of  a  spheric  segment  be  2,  and  radius 
of  base  6,  then 

(6a  X  3  +  2a)  X  2  X  -5236  =  117-2864  =  content 

7.  For  the  solidity  of  a  steroid. 
Revolving  axis  3X  fixed  axis  x  -5236  =  content. 
NOTE. — If  the  spheroid  revolVe  round  the  greater  axis, 
it  is  said  to  be  oblate ;  if  round  the  lesser,  oblong. 

Ex. — The  two  axes  of  a  spheroid  are  24  and  18;  there- 
fore, 

24  SX  18  X  -5236  =  5428-56  =  content  if  it  be  oblate. 
18  3X  24  X  -5236  =  4071-5  =  content  if  it  be  oblong. 

8.  For  the  solidity  of  a  parabolic  conoid. 

Area  of  base  X  half  the  height  =  content. 
Ex. — The  height  being  18,  and  the  diameter  of  base  24, 
then  the  area  of  the  base  therefore  is  452-39 ;  hence 
452-39  X  9  =  4071-51  the  content. 

9.  For  the  frustum  of  a  cone  or  pyramid. 
(perim.  of  one  end  -}-  perim.  of  the  other  end)  x  slant  height 

~2~ 
=  surface. 

Ex. — In  the  frustum  of  a  triangular  pyramid  the  peri- 
meter of  one  end  is  25,  that  of  the  other  36,  and  the  slant 
height  is  10  ;  therefore, 

(25  +  36)  x  10 

— =  305  =  the  surface. 
!c 

^(area  of  one  end  +  ar.  of  other)  +  area  of  one  end  +  ar.  of  other 

3T~ 

X  height  =  content. 
Ex. — A  log  of  wood  is  20  feet  long  ;  its  ends  are  squares, 


MENSURATION.  107 

•f  which  the  sides  are  respectively  12  and  16  inches  ;  there- 
fore, 

-  <  _  con(en, 


TIMBER  MEASURE. 

EXAMPLES  of  timber  measuring  have  already  been  given  in 
the  department  allotted  to  arithmetic,  but  it  is  necessary  to 
be  here  somewhat  more  particular.  The  surface  of  a  plank 
is  found : — 

1st.  By  multiplying  the  length  by  the  breadth.  When 
the  board  tapers  gradually,  add  the  breadth  at  both  ends 
together,  and»  take  the  half  of  this  sum  for  the  mean 
breadth. 

2d.  By  the  sliding  rule. — Set  the  length  in  inches  on 
B  to  12  on  A,  and  against  the  length  in  feet  on  B  will  be 
the  area  in  square  feet  and  decimals  on  A. 

Ex.  —  A  board  is  12  feet  6  inches  long  and  1  foot  3 
inches  broad ;  hence, 

12     :     6 
1     :     3 

T2 :     6 
3:1:6 


15     :     7     :     6 

1st.  For  the  content  of  squared  timber,  length  x  mean 
breadth  X  mean  thickness  =  content. 

2rf.  By  the  sliding  rule. — Find  the  mean  proportional 
between  the  breadth  and  thickness,  then  set  the  length  on 
C  to  12  on  D,  and  against  the  mean  proportional  on  D  the 
solid  content  on  C.  If  the  mean  proportional  be  in  feet, 
reduce  to  inches. 

Ex. — A  log  is  24  feet  long,  the  mean  depth  and  breadth 
being  each  13  inches. 

1   :   1 
1   :   1 


1:2:1 
24 

28  :  2  :  0 


10S  MENSURATION. 

For  round  timber. — 1st.  Take  one-fourth  of  the  mean 
girth  and  square  it,  this  multiplied  by  the  length  will  give 
die  content. 

2d.  By  the  sliding  rule. — Set  the  length  in  feet  on  C 
to  12  on  D,  then  against  the  quarter  girth  in  inches  on  D, 
will  be  the  content  on  C. 

This  gives  no  allowance  for  bark,  but  there  is  usually  a 
deduction  made  of  about  an  inch  to  the  foot  of  quarter  girth. 
The  rule  given  above  gives  the  customary,  but  not  the  true 
ontent ;  the  following  gives  the  true  content. 

One-fifth  of  the  girth  squared  and  multiplied  by  twice  the 
length  =  content. 

Ex. — The  mean  girth  of  a  tree  being  5  feet  8  inches,  and 
its  length  18  feet,  the  two  rules  will  apply  as  below: — 
4)  5  :  8  (1   :  5  5)  5  :  8  (1   :   1   :  7 

1   ;  5  1   ;   1   ;  7 

2:0:1  T~:  3  :  4  :  6 

18  36 


36  :   1   :  6  46  :   1   :  6 

Trees  very  seldom  have  an  equal  girth  throughout,  one 
end  being  generally  much  smaller  than  the  other :  the  girth 
taken  above  is  the  mean  girth ;  that  is  to  say,  the  girths  of 
both  ends  added  together,  and  their  sum  halved  for  the 
mean  girth.  It  is  to  be  observed,  however,  that,  if  the 
difference  of  the  girths  is  great,  it  will  be  best  to  find  the 
content  of  the  tree  as  if  it  $vere  a  conic  frustum. — The 
method  of  using  the  sliding  rule  in  the  measurement  of  tim- 
ber has  been  given  before. 


ARTIFICERS'  WORK. 

ARTIFICERS  compute  the  contents  of  their  works  by 
several  different  measures ;  as,  glazing  and  masonry  by  the 
foot;  painting,  plastering,  paving,  &c.,  by  the  yard  of  9 
square  feet;  flooring,  partitioning,  roofing,  tiling,  &c.,  by 
the  square  of  100  square  feet;  and  brickwork,  either  by 
a  yard  of  9  square  feet,  or  by  the  perch,  or  square  rod  or 
pole,  containing  2721  square  feet,  or  30.1  square  yards, 
being  the  square  of  the  rod  or  pole  of  16s  feet  of  5^  yards 
long.  As  this  number  272.j  is  troublesome  to  divide  by, 
the  1  is  often  omitted  in  practice,  and  the  content  in  feet 
divided  only  by  the  272.  But  when  the  exact  divisor 


ARTIFICERS'  WORK.  109 

272]  is  to  be  used,  it  will  be  easier  to  multiply  the  feet  by 
4,  and  then  divide  successively  by  9,  11,  and  11.  Also  to 
divide  square  yards  by  30],  first  multiply  them  by  4,  and 
then  divide  twice  by  11. 

BRICKLAYERS'  WORK. — Brickwork  is  estimated  at  the 
rate  of  a  brick  and  a  half  thick.  So  that,  if  a  wall  be  more 
or  less  than  this  standard  thickness,  it  must  be  reduced  to 
it,  as  follows  : — Multiply  the  superficial  content  of  the  wall 
by  the  number  of  half  bricks  in  the  thickness,  and  divide 
the  product  by  3.  The  dimensions  of  a  building  are  usu- 
ally taken  by  measuring  half  round  on  the  outside,  and  half 
round  on  the  inside ;  the  sum  of  these  two  gives  the  com- 
pass of  the  wall, — to  be  multiplied  by  the  height,  for  the 
content  of  the  materials.  Chimneys  are  by  some  measured 
as  if  they  were  solid,  deducting  only  the  vacuity  from  the 
hearth  to  the  mantel,  on  account  of  the  trouble  of  them. 
And  by  others  they  are  girt  or  measured  round  for  their 
breadth,  and  the  height  of  the  story  is  their  height,  taking 
the  depth  of  the  jambs  for  their  thickness.  And  in  this 
case,  no  deduction  is  made  for  the  vacuity  from  the  floor 
to  the  mantel-tree,  because  of  the  gathering  of  the  breast 
and  wings,  to  make  room  for  the  hearth  in  the  next  story. 
To  measure  the  chimney  shafts,  which  appear  above  the 
building,  gird  them  about  with  a  line  for  the  breadth,  to 
multiply  by  their  height.  And  account  their  thickness 
half  a  brick  more  than  it  really  is,  in  consideration  of  the 
plastering  and  scaffolding.  All  windows,  doors,  &c.,  are 
to  be  deducted  out  of  the  contents  of  the  walls  in  which 
they  are  placed.  But  this  deduction  is  made  only  with 
regard  to  materials ;  for  the  whole  measure  is  taken  for 
workmanship,  and  that  all  outside  measure  too,  namely, 
measuring  quite  round  theoutside  of  the  building,  being 
in  consideration  of  the  trouble  of  the  returns  or  angles. 
There  are  also  some  other  allowances,  such  as  double 
measure  for  feathered  ffahle  ends,  &c. 
•  Ex. — The  end  wall  of  a  house  is  28  feet  10  inches  long", 
"ai\'\  55  tVet  8  inches  high,  to  the  eaves:  20  feet  high  is  2' 
bucks  thick,  other  20  feet  high  is  2  bricks  thick,  and  the 
remaining  15  feet  8  inches  is  H  brick  thick;  above  which 
is,  a  triangular  gable,  1  brick  thick,  and  which  rises  42 
courses  of  bricks,  of  which  every  4  courses  make  a  foot. 
What  is  the  whole  content  in  standard  measure  ? 

Ans.  253-626  yards. 
10 


110  MENSURATION 

MASONS'  WORK. — To  masonry  belong  all  sorts  of  stone- 
work ;  and  the  measure  made  use  of  is  a  foot,  either  super- 
ficial or  solid.  AValls,  columns,  blocks  of  stone  or  marble, 
&c.,  are  measured  by  the  cubic  foot;  and  pavements, 
slabs,  chimney-pieces,  &c.,  by  the  superficial  or  square 
foot.  Cubic  or  solid  measure  is  used  for  the  materials, 
and  square  measure  for  the  workmanship.  In  the  solid 
measure,  the  true  length,  breadth  and  thickness,  are  taken, 
and  multiplied  continually  together.  In  the  superficial, 
there  must  be  taken  the  length  and  breadth  of  every  part 
of  the  projection,  which  is  seen  without  the  general  upright 
face  of  the  building. 

Ex. — In  a  chimney-piece,  suppose  the 
Length  of  the  mantel  and  slab,  each  4  feet  6  inches ; 

Breadth  of  both  together, 3          2 

Length  of  each  jamb, 4          4 

Breadth  of  both  together, 1          9 

Required  the  superficial  content.        Ans.  21  feet,  10  inch. 

CARPENTERS'  AND  JOINERS'  WORK. — To  this  branch 
belongs  all  the  wood-work  of  a  house,  such  as  flooring, 
partitioning,  roofing,  &c.  Large  and  plain  articles  are 
usually  measured  by  the  square  foot  or  yard,  &c.,  but 
enriched  mouldings,  and  some  other  articles,  are  often 
estimated  by  running  or  lineal  measures,  and  some  things 
are  rated  by  the  piece. 

In  measuring  of  joists,  it  is  to  be  observed,  that  only  one 
of  their  dimensions  is  the  same  with  that  of  the  floor ;  for 
the  other  exceeds  the  length  of  the  room  by  the  thickness 
of  the  wall  and  |  of  the  same,  because  each  end  is  let  into 
the  wall  about  f  of  its  thickness. 

No  deductions  are  made  for  hearths,  on  account  of  the 
additional  trouble  and  waste  of  Viaterials. 

Partitions  are  measured  from  wall  to  wall  for  one  dimen- 
sion, and  from  floor  to  floor,  as  far  as  they  extend,  for  the  other. 

No  deduction  is  made  for  door-ways,  on  account  of  the 
trouble  of  framing  them. 

In  measuring  of  joiners'  work,  the  string  is  made  to  ply 
close  to  every  part  of  the  work  over  which  it  passes. 

The  measure  for  centering  for  cellars  is  found  by  making 
a  string  pass  over  the  surface  of  the  arch  for  the  breadth, 
and  taking  the  length  of  the  cellar  for  the  length ;  but  in 
groin  centering,  it  is  usual  to  allow  double  measure,  on 
account  of  their  extraordinary  trouble. 


ARTIFICERS'  WORK.  Ill 

In  roofing,  the  length  of  the  house  in  the  inside,  to- 
gether with  ~  of  the  thickness  of  one  gable,  is  to  be  con- 
sidered as  the  length  ;  and  the  breadth  is  equal  to  double 
the  length  of  a  string  which  is  stretched  from  the  ridge 
down  the  rnfic-r,  and  along  the  eaves-board,  till  it  meets  with 
the  top  of  the  wall. 

For  staircases,  take  the  breadth  of  all  the  steps,  by  making 
a  line  ply  close  over  them,  from  the  top  to  the  bottom,  and 
multiply  the  length  of  this  line  by  the  length  of  a  step,  for 
the  whole  area. — By  the  length  of  a  step  is  meant  the 
length  of  the  front  and  the  returns  at  the  two  ends  ;  and 
by  the  breadth,  is  to  be  understood  the  girth  of  its  two  outer 
surfaces,,  or  the  tread  and  riser. 

For  the  balustrade,  take  the  whole  length  of  the  upper 
part  of  the  hand-rail,  and  girt  oVer  its  end  till  it  meet  the 
top  of  the  newel  post,  for  the  length ;  and  twice  the  length 
of  the  baluster  upon  the  landing,  with  the  girth  of  the  hand- 
rail, for  the  breadth. 

For  wainscoting,  take  the  compass  of  the  room  for  the 
length  ;  and  the  height  from  the  floor  to  the  ceiling,  making 
the  string  ply  close  into  alb  the  mouldings  for  the  breadth. T— 
Out  of  this  must  be  made  deductions  for  windows,  doors, 
and  chimneys,  &c.,  but  workmanship  is  counted  for  the 
whole,  on  account  of  the  extraordinary  trouble. 

For  doors,  it  is  usual  to  allow  for  their  thickness,  by  add- 
ing it  unto  both  the  dimensions  of  length  and  breadth,  and 
then  to  multiply  them  together  for  the  area.  If  the  door 
be  paneled  on  both  sides,  take  double  its  measure  for  the 
workmanship ;  but.  if  the  one  side  only  be  paneled,  take 
the  area  and  its  half  for  the  workmanship. — For  the  sur- 
rounding architrave,  sfird  it  about  the  outermost  parts  for 
its  length  ;  and  measure  over  it,  as  far  as  it  can  be  seen 
when  the  door  is  open,  for  the  breadth. 

Window-shutters,  bases,  &c.,  are  measured  in  the  same 
manner. 

In  the  measuring  of  roofing  for  workmanship  alone, 
Holes  for  chimney-shafts  and  skylights  are  generally  de- 
dusted.  But  in  measuring  for  work  and  materials,  they 
commonly  measure  in  all  skylights,  luthern-lights,  and 
holes  for  the  chimney-shafts,  on  account  of  their  trouble 
and  waste  of  materials. 

Ex. — To  how  much,  at  Qs.  per  square  yard,  amounts  the 
wainscoting  of  a  room  ;  the  height,  Baking  in  the  cornice 


£  MENSURATION. 

and  mouldings,  being  12  feet  6  inches,  and  the  whole  com- 
pass 83  feet  8  inches  ;  also  three  window- shutters  are  each 
7  feet  8  inches  by  3  feet  6  inches,  and  the  door  7  feet  by  3 
feet  6  inches  ;  the  door  and  shutters,  being  worked  on  both 
sides,  are  reckoned  work  and  half  work  ? 

Ans.  £36,  12s.  2%d. 

SLATERS'  AND  TILERS'  WORK. — In  these  articles,  the 
content  of  a  roof  is  found  by  multiplying  the  length  of  the 
ridge  by  the  girth  over  from  eaves  to  eaves  ;  making  allow- 
ance in  this  girth  for  the  double  row  of  slates  at  the  bottom, 
or  for  how  much  one  row  of  slates  or  tiles  is  laid  over  an- 
other. When  the  roof  is  of  a  true  pitch,  that  is,  forming 
a  right  angle  at  top,  then  the  breadth  of  the  building  with 
its  half  added,  is  the  girth  over  both  sides.  In  angles 
formed  in  a  roof,  running  from  the  ridge  to  the  eaves,  when 
(he  angle  bends  inwards,  it  is  called  a  valley;  but  when 
outwards,  it  is  called  a  hip.  Deductions  are  made  for 
chimney-shafts  or  window-holes. 

Ex. — To  how  much  amounts  the  tiling  of  a  house,  at 
25s.  6d.  per  square  ;  the  length  being  43  feet  10  inches, 
and  the  breadth  on  the  flat  27  feet  5  inches,  also  the  eaves 
projecting  16  inches  on  each  side,  and  the  roof  of  a  true 
pitch?  £24,  9s.  5|rf. 

PLASTERERS'  WORK. — Plasterers'  work  is  of  two  kinds, 
namely,  ceiling — which  is  plastering  upon  laths — and  ren- 
''ering,  which  is  plastering  upon  walls  ;  which  are  measured 
separately. 

The  contents  are  estimated  either  by  the  foot  or  yard,  or 
square  of  100  feet.  Enriched  mouldings,  <fcc.,  are  rated  by 
running  or  lineal  measure. 

Deductions  are  to  be  made  for  chimneys,  doors,  win-  • 
dows,  &c.     But  the  windows  are  seldom  deducted,  as  the 
plastered  returns  at  the  top  and  sides  are  allowed  to  com 
pensate  for  the  window  opening. 

Ex. — Required  the  quantity  of  plastering  in  a- room,  the 
length  being  14  feet  5  inches,  breadth  13  feet  2  inches,  and 
height  9  feet  3  inches  to  the  under  side  of  the  cornice, 
which  girts  85  inches,  and  projects  5  inches  from  the  wall 
on  the  upper  part  next  the  ceiling — deducting  only  for  a 
door  7  feet  by  4. 

Ans.  53  yards  5  feet  3  inches  of  rendering, 
18  5          6  of  ceiling, 

39-         O  of  cornice. 


ARTIFICERS'  WORK.  113 

PAINTERS'  WORK. — Painters'  work  is  computed  in  square 
yards.  Every  part  is  measured  where  the  colour  lies  ;  and 
the  measuring  line  is  forced  into  all  the  mouldings  and 
corners. 

Windows  are  done  at  so.  much  apiece.  And  it  is  usual 
to  illow  double  measuYe  for  carved  mouldings,  &c. 

Ex. — What  costs  the  painting  of  a  room  at  6d.  per  yard  ; 
its  length  being  24  feet  6  inches,  its  breadth  16  feet  3 
inches,  and  height  12  feet  9  inches ;  also  the  door  is  7  feet 
by  3  feet  6  inches,  and  the  window-shutters  to  two  windows 
each  7  feet  9  inches  by  3  feet  6  inches,  but  the  breaks  of 
the  windows  themselves  are  8  feet  6  inches  high,  and  1  fo.ot 
3  inches  deep — deducting  the  fire-place  of  5  feet  by  5  feet 
6  inches  ?  Ans.  dB3,  3s.  10|rf. 

GLAZIERS'  WORK. — Glaziers  take  their  dimensions  either 
in  feet,  inches,  and  parts  ;  or  feet,  tenths,  and  hundredths. 
And  they  compute  their  work  in  square  feet. 

In  taking  the  length  and  breadth  of  a  window,  the  cross 
bars  between  the  squares  are  included.  Also,  windows  of 
round  or  oval  forms  are  measured  as  square,  measuring 
them  to  their  greatest  length  and  breadth,  on  account  of  the 
waste  in  cutting  the  glass. 

Ex. — Required  the  expense  of  glazing  the  windows  of  a 
house  at  134.  a  foot;  there  being  three  stories,  and  three 
windows  in  each  story. 

The  height  of  the  lower  tier  is  7  feet  9  inches, 

'..     of  the  middle  6         6 

of  the  upper  5         3| 

and  of  an  oval  window  over  the  door  1        10 k 

the  common  breadth  of  all  the  windows  being  3  feet  9 

inches.  Ans.   £12,  Is.  8-irf. 

PAVERS'  WORK. — Pavers'  work  is  done  by  the  square 
yard.  And  the  content  is  found  by  multiplying  the  length 
by  the  breadth. 

Ex. — What  will  be  the  expense  of  paving  a  rectangular 
courtyard, "whose  length  is  63  feet,  and  breadth  45  feet ; 
in  which  there  is  laid  a  footpath  of  5  feet  3  inches  broad, 
running  the  whole  length,  with  broad  stones,  at  3s.  a  yard— 
tbj£  rest  being  paved  with  pebbles,  at  Is.  67.  per  ynrd  ? 

Ans.  £40,  5«.10£</. 

TLTTMBERS'  WORK. — Plumbers'  work  is  rated  at  so  much 
a  pound,  or  else  by  the  hundred  weight,  of  112  pounds. 
Sheet  lead  used  in  roofing,  sputtering,  &c.,  is  from  7  to  13 
10* 


114  MENSURATION. 

lb.  to  the  square  foot.     And  a  pipe  of  an  inch  bore  is  com- 
monly 13  to  14  lb.  to  the  yard  in  length. 

Ex. — What  cost  the  covering  and  guttering  a  roof  with 
lead,  at  19s.  the  cwt. ;  the  length  of  the  roof  being  43  feet, 
and  breadth  or  girth  over  it  32  feet — the  guttering  60  feet 
long,  and  2  feet  wide,  the  former  9  lb.,  and  the  latter  8  lb 
to  the  square  foot  ?  Ans  £113,  3s.  Sid 


MECHANICS. 


DEFINITIONS. 

1.  A  BODY  is  any  quantity  of  matter  collected  together. 

2.  Whatever  communicates,  or  has  a  tendency  to  com- 
miniciite  motion  to  a  body,  is  called  a  force. 

3.  That  department  of  knowledge  which  comprehends  a 
statement  of  the  effects  of  forces  on  bodies,  is  called  Mecha 
nics.     If  a  body  be  put  in  motion  by  the  action  of  one  or 
more  forces,  the  consideration  of  the  circumstances  of  this 
body  belongs  to  that  branch  of  Mechanics  called  Dynamics; 
but  if  two  or  more  forces  act  on  a  body  in  such  a  way  that 
they  destroy  each  other's  effects,  and  the  body  remains  at 
rest,  or  in  equilibrium,   the  consideration  of  the  circum- 
stances of  a  body,  in  this  case,  belongs  to  that  department 
of  Mechanics  called  Statics. 

4.  The  density  of  matter,  is  the  quantity  of  matter  con- 
tained in  any  body  compared  with  its  bulk.     Thus  lead  is 
denser  than  cork. 

5.  The  weight  of  a  body,  is  its  quantity  of  matter,  with- 
out regard  to  its  bulk. 

6.  When  we  speak  of  some  given  space,  which  a  moving 
body  passes  over  in  a  given  time,  we  speak  of  the  velocity 
of  the  body.    If  a  body  moves  over  one  foot  of  space  in  one 
second  of  time,  it  is  said  to  have  a  velocity  of  one  foot  in 
the   second ;    and   its  velocity  would  be  increased  to  the 
double,  if  it  passed  over  two  feet  in  one  second  of  time. 

7.  If,  while  the  body  is  in  motion,  the  velocity  continues 
the  same,  the  body  is  said  to  have  a  uniform  motion ;  but 
if,  while  the  body  moves  onward,  the  velocity  continually 
increases,  it  is  said  to  have  an  accelerated  motion  ;  and,  on 
the/other  hand,  if  during  the  progress  of  the  body  in  motion, 
the  velocity  continually  decreases,  the  body  is  said  to  have* 
a  retarded  motion. ' 

8.  The  quantity  of  matter  in  a  moving  body,  multiplied 
by  the  velocity  with  which  it  moves,  is  called  the  quantity 
of  motion,  or  momentum  of  the  body. 

115 


116 


MECHANICS. 


9.  Gravity  is  that  force  by  which  all  bodies  endeavour  to 
descend  towards  the  centre  of  the  earth. 


AXIOMS,  OR  PLAIN  TRUTHS. 

IF  a  body  be  at  rest,  it  will  remain  at  rest ;  and  if  in  mo- 
tion, it  will  continue  that  motion,  uniformly  in  a  straight 
line,  if  it  be  not  disturbed  by  the  action  of  some  external 
cause. 

The  change  of  motion  takes  place  in  the  direction  in 
which  the  moving  force  acts,  and  is  proportional  to  it. 

The  action  and  reaction  of  bodies  upon  one  another,  are 
equal. 

LAWS    OF    MOTION. 

Uniform  motion  is  caused  by  the  action  of  some  force. 

by  one  impulse,  on  the  body  : — and  if 

b  signify  the  quantity  of  matter  to  be  moved, 
f  the  force  which  caused  the  body's  motion, 
v  the  velocity  with  which  the  body  moves, 
m  the  momentum  of  the  body  in  motion, 
s  the  space  passed  over  by  the  moving  body, 
t  the  time  of  describing  that  space  ; 

and  if  b  =  3,  m  =  6,  v  =  2,/  =  -6,  s  =  4,  t  =  2  :    then 

the  figures  in  the  examples  will  show  the  application  of  the 

theorems. 


b: 

m 

s  : 

1 

v  : 

t  : 

m 

v 

m  : 

t  X 

m 

T  : 

s 

THEOREMS. 

fxt 

3: 
6  : 
6  : 
4  : 
2: 
2: 

6 

EXAMPLES. 

6       6x2      6X2 

V 

b  X  v 
b   x  v 

tx 

V  ' 

s 
bx 

s 

s 

2 

6 

6 

2 

6 
"3 
4 

'    2    ' 
:  3  x2 

:  3  X2 
X2-2 

4              4 
3X4 

t 
b  X 

s 

3 

2 
X4 

t 
m     t 

x/ 

X 

2 

6      2X6 

s 

7 

*  X  b 

b 

f 
b 

s  X  b 

b 

3 

46 

rt*      *         O 

-^              O 

4x3 

3 
4x3 

v 

m 

f 

2 

'       6 

6 

MOTION.  117 

OF    ACCELERATED    MOTION. 

If  the  moving  force  continues  to  act  all  the  while  that  the 
body  is  in  motion,  then  that  motion  will  be  uniformly  ac- 
celerated :  such  is  the  case  with  bodies  falling  to  the  earth, 
as  the  force  of  gravity  acts  constantly.  Now,  it  has  been 
found  by  experiment,  that  a  body  falling  through  free  space, 
in  the  latitude  of  London,  will,  by  the  force  of  gravity,  fall 
through  16*095  feet  in  the  rirst  second  of  time  ;  and  as  forces 
are  measured  by  the  effects  ihey  produce,  this  16'095  may 
be  taken  as  the  measure  of  the  force  of  oravity  ;  and  as  this 
quantity  does  not  differ  materially  from  16  feet",  we  shall 
neglect  the  fraction  '095  in  our  calculation  of  the  circum- 
stances of  falling  bodies. 

The  subjects  of  consideration  here  are,  the  time  that  the 
falling  body  is  in  motion,  the  spar*:  it  falls  through  in  that 
tinie.  and  the  velocity  which  it  has  acquired  in  falling 
through  that  space,  or  that  velocity  with  which  it  would 
continue  to  move,  supposing  gravity  to  cease  its  action,  and 
the  motion  of  the  body  becoming  uniform. 

The  time  is  always  supposed  to  be  taken  in  seconds,  and 
the  space  in  feet. 

The  velocity  acquired  =  32  x  time  of  falling:, 

or  =x/(64  x  space  fallen  through) 

_,,       .          rrii-                      the  velocitv  acquired 
The  time  of  falling  = ~ • 

040 

I/the  space  fallen  through  \ 
%'  16  ' 

the  velocitv  acquired  * 
Tne  space  fallen  through  = —        — ^ — 

or  =  time,  2  x  16. 

Ex. — If  a  body  falls  through  100  feet,  then 
-v/(64  x  100)  =  80  =  the  velocity  acquired 

80 

— r=  2-^v  =  2-5  =  the  time  of  falling. 

If  /the  space  described  be  64  feet,  then 

Iff*  4  \ 

I- — -  =  2  =  the  time  of  falling, 

32  x  2  =  64  =  the  velocity  acquired. 
If  the  space  descended  be  400,  then 


118  MECHANICS. 

v/(400  x  64)  =  160  =  the  velocity  acquired, 
•L-^— -  =  5  =  the  time  of  falling. 

o£ 

If  the  times  be  as  1,  2,  3,    4,     5,  &c. 

The  velocities  will  be  as     1,  2,  3,    4,     5,  &c. 
And  the  spaces  as  1,  4,  9,  16,  25,  &c. 

The  space  for  each  time  as  1,  3,  5,    7,    9,  &c. 


COLLISION  OF  BODIES. 

IF  two  bodies,  A  and  B,  in  motion,  weigh  respectively  5 
and  3  Ibs.,  and  their  velocities  respect- 
ively 3  and  2  before  they  strike,  -  -  -  -  -  . 
then  will  3  X  5  be  the  momentum  of 
A,  and  2x3  that  of  B,  before  the  stroke;  also,  5  -f  3  = 
8  is  the  sum  of  their  weights  ;  then,  1st.  If  the  bodies  move 
the  same  way,  the  quotient  arising  from  the  division  of  the 
sum  of  the  momentums  of  the  two  bodies,  by  the  sum  of 
their  weights,  will  give  the  common  velocity  of  the  two 
bodies  after  the  stroke.  2d.  If  the  bodies  move  contrary 
ways,  then  the  quotient  arising  from  the  division  of  the 
difference  of  their  momentums,  by  the  sum  of  their 
weights,  will  give  the  common  velocity  after  the  stroke. 
3d.  If  one  of  the  bodies  be  at  rest,  then  the  quotient  of  the 
momentum  of  the  other  body,  divided  by  the  sum  of  the 
weights  of  the  two  bodies,  will  give  the  common  velocity 
after  the  stroke.  Hence,  assuming  the  numbers  given  above, 

1  e       (       f* 

we  have,  in  the  first  case,  —         —  =  2|  ;   in  the  second 

8 

-  -  =  1|  ;  and  in  the  third  —  -  =  1§,  as  the  common 
8  8 

velocity  after  the  stroke. 

When  the  bodies  are  perfectly  elastic,  the  theorems  be- 
come more  complicated. 

If  the  weight  of  the  one  body  be  A,  and  the  velocity  V  ; 
the  weight  of  the  other  body  B,  and  its  velocity  v  :  then, 

1st.  If  the  bodies  move  in  the  same  direction  before  the 
stroke, 

(2Bxt>)—  (A-BxV)  ,  .     .       . 

-  -  =  the  velocity  of  A  after  the  stroke. 
A-f-r> 


_  the  velocity  of  B  after  the  stroke. 
A+L* 


PARALLELOGRAM    OF    FORCES.  19 

2</.  If  B  move  in  the  contrary  direction  to  A  before  the 
stroke, 

(A-B)xV-2xBxV 

i — — =  velocity  of  A  after  the  stroke. 

(A— B)  xy+2+AxV 

— -— -       =  velocity  of  B  after  the  stroke. 

3d.  If  the  body  B  had  been  at  rest  before  it  was  struck 
by  A,  then 

A  I> 

X  V  =  the  velocity  of  A  after  the  stroke. 
X  V  =  the  velocity  of  B  after  the  stroke. 


A  +  B 
2  —  A 


A  x  B 

Ex. — If  the  weight  of  an  elastic  body  A  be  6  Ibs.,  and 
its  velocity  4,  and  the  weight  of  another  body  B  be  4  Ibs., 
and  its  velocity  2;  then  we  have  these  results:  in  the  first 
case, 

(2x4x2)  +  (6— 4x4)  ,  .     ,       . 

—  =  -8  =  velocity  of  A  alter  the  stroke. 
6  +  4 

(2x6x4)  +  (6— 4X2) 

i — ± =  5-2  velocity  of  B  after  the  stroke. 

The  sum  of  these  two  velocities,  viz.  5-2  and  -8  =  6,  which 
was  the  sum  of  the  velocities  2  and  4  before  the  stroke ; 
and  this  is  a  general  law. — The  reader  may  exercise  him- 
self with  the  rules  for  the  other  cases. 

It  is  to  be  observed,  that  when  non-elastic  bodies,  that  is, 
bodies  which  have  no  spring,  strike,  they  will  both  move 
in  the  direction  of  the  motion  of  that  body  which  has  the 
greater  momentum  ;  but  if  they  are  elastic,  they  will  recoil 
after  the  stroke,  and  move  contrary  ways. 


THE  COMPOSITION  AND  RESOLUTION  OF  FORCES. 

j  IF  a  body  be  acted  upon 
By  two  forces,  one  of  which 
would  cause  it  to  move  from 
A  to  B  in  any  given  time, 
and  the  other  would  cause 
it  to  move  from  A  to  C  m 


12!  MECHANICS 

the  same  time ;  then  if  these  forces  act  upon  the  body  at 
one  instant,  it  will  move  in  neither  .of  the  lines  AB,  AC, 
but  in  the  line  AD,  which  is  the  diagonal  of  the  parallelo- 
gram of  which  the  two  lines  AB  and  AC  are  containing 
sides ;  and  by  the  action  of  the  two  forces,  the  body  will  be 
found  at  D,  at  the  end  of  the  time  that  it  would  have,  been 
found  at  B  or  C,  by  the  action  of  either  of  the  forces  singly 
This  important  fact  in  mechanical  science,  is  usually  called 
the  parallelogram  of  forces.  From  this  statement  it  will 
be  seen,  that  if  we  have  the  quantity  and  direction  of  any 
two  forces  urging  a  body  at  the  same  instant,  we  can  find 
the  resulting  motion,  both  in  quantity  and  direction. 

It  will  not  be  difficult  to  understand,  that  if  the  two  forces 
which  act  upon  a  body,  act  not  at  an  angle,  but  in  the  same 
straight  line,  and  in  contrary  directions,  the  resulting 
motion  will  be  in  that  straight  line,  and  in  the  direction  of 
the  greater  force  ;  but  if  the  forces  be  equal,  the  body  will 
remain  at  rest.  If,  while  a  body  A  is  urged  by  a  force  in 
the  direction  AB,  which  would  carry  it  to  A,  it  be  acted 
on  by  another  force  in  the  direction  AC  which  would  carry 
it  to  C,  and  a  third  force  in  the  direction  DA,  which  would 
carry  it  over  a  space  as  great  as  that  from  D  to  A,  these 
being  the  sides  and  diagonals  of  a  parallelogram,  the  body 
A  will  remain  at  rest.  Also,  if  a  body  A  has  a  tendency  to 
move  in  the  direction  AB,  but  is  counteracted  by  a  force 
DA, — and  if  we  wish  to  keep  the  body  A  from  moving, 
altogether,  we  must  apply  another  force  AC,  forming  the 
other  side  of  the  parallelogram  of  which  AB  is  one  side  and 
AD  the  diagonal. 

If  there  be  three  forces  acting  on  a  body  at  the  same  time, 
make  the  sides  of  a  parallelogram  represent  any  two  of 
them  ;  then  the  diagonal  of  this  parallelogram,  together 
with  the  third  force  as  the  two  sides  of  another  parallelo- 
gram, will  give  a  diagonal  which  will  be  the  result  of  the 
three  forces  acting*  at  once  on  the  body. 

If  the  two  forces  which  urge  the  body,  both  produce  a 
uniform  motion,  the  resulting  motion  will  be  in  a  straight 
line;  tout  if  one  of  them  act  by  impulse,  which  would  pro- 
duce a  uniform  motion,  and  the  other  act  constantly  so  as 
to  produce  an  accelerated  motion,  the  resulting  motion  will 
be  in  a  curve.  Thus,  if  the  ball  of  a  cannon  were  sent  in 
a  horizontal  direction,  it  would  never  deviate  from  this 
straight  line  unless  acted  on  by  some  external  force.  The 


THE    LEVER.  12} 

force  of  gravity  acts  on  the  body  constantly,  so  as  to  draw 
it  to  the  earth,  by  a  uniformly  accelerated  motion ;  and  the 
result  is,  that  the  ball  will  move  in  a  curve,  and  this  curve 
may  bo  easily  shown  to  be  that  of  the  parabola.  The  re- 
sistance of  the  air  being  taken  into  account  together  with 
these  circumstances,  constitute  the  basis  of  the  science  of 
gunnery. 

We  shall  give  a  simple  example,  to  show  the  application 
of  the  former  part  of  this  subject.  One  force  will  cause 
the  body  A  to  move  20  miles  in  a  day,  and  another,  acting 
at  right  angles,  will  cause  it  to  move  18  miles  a  day;  draw 
these  lines  20  and  18  from  the  line  of  lines  on  the  sector, 
as  the  sides  AB,  AC,  of  a  parallelogram,  and  complete  it: 
draw  the  diagonal,  then  measure  it,  and  it  will  be  found 
to  be  26'9,  the  resulting  motion ;  and  the  angle  being 
measured,  will  give  the  direction. — There  are  other  methods 
of  doing  this  by  calculation,  but  this  is  simple,  and  is  suffi- 
cient to  show  the  principle. 


MECHANICAL  POWERS. 

1.  A  MACHINE  is  any  instrument  employed  to  regulate 
motion,  so  as  to  save  either  time  or  force.    No  instrument 
can  be  employed  by  man  so  as  to  save  both  time  and  force ; 
for  it  is  a  maxim  in  mechanics,  that  whatever  we  gain  in 
the  on"  of  these  two,  must  be  at  the  expense  of  the  other. 

2.  The  simple  machines,  or  those  of  which  all  others  are 
constructed,  are  usually  reckoned  six  :  the  lever,  the  wheel 
and  axle,  the  pulley,  the  inclined  plane,  the  wedge,  and 
the  screw.     To  these  the  funicular  machine  is  sometimes 
added. 

3.  The  weight  signifies  the  body  to  be  moved,  or  th« 
resistance  to  be  overcome ;   and  the  power  is  the  force  em 
ployed  to  overcome  that  resistance,  or  move  that  body 
They  are  frequently  represented  by  the  first  letters  of  theii 
.names,  W  and  P. 

THE   LEVER. 

4.  A  LEVER  is  an  inflexible  bar,  either  straight  or  bent, 
supposed  capable  of  turning  round  a  fixed  point,  called  the 
fulcrum 

1] 


122  MECHANICS. 

According  to  tho  relative  positions  of  tho  weight,  power 
and  fulcrum,  on  the  lever,  it  is  said  to  be  of  three  kinds, 
viz.  when  the  fulcrum  is  somewhere  betwixt  the  weight  and 
power,  it  is  of  the  first  kind  ;  when  the  weight  is  between 
the  power  and  the  fulcrum,  it  is  of  the  second  kind ;  and 
when  the  power  is  between  the  weight  and  the  fulcrum,  it 
is  of  the  third  kind :  thus, 

5.  1st. - 


6.  3d. 

7.  3d. 


8.  In  the  first  and  second  kinds  there  is  an  advantage  of 
power,  but  a  proportionate  loss  of  velocity ;  and  in  the  third 
kind,  there  is  an  advantage  in  velocity,  but  a  loss  of  power. 

9.  When  the  weight  X  its  distance  from  the  fulcrum  = 
the  power  x  its  distance  from  the  fulcrum,  then  the  lever 
will  be  at  rest,  or  in  equilibrio  ;   but  if  one  of  these  pro- 
ducts be  greater  than  the  other,  the  lever  will  turn  round 
the  fulcrum  in  the  direction  of  that  side  whose  product  is 
the  greater. 

10.  In  all  the  three  kinds  of  levers,  any  of  these  quanti- 
ties, the  weight  or  its  distance  from  the  fulcrum,  or,  the 
power  or  its  distance  from  the  fulcrum,  may  be  found  from 
the  rest,  such,  that  when  applied  to  the  lever,  it  will  remain 
at  rest,  or  the  weight  and  power  will  balance  each  other. 

weight  X  its  dist.  from  fulc. 

11.  — -re2 7 —       —„ TT-J =  power. 

dist.  of  power  from  fulc. 

power  x  its  dist.  from  fulc. 
dist.  of  weight  from  fulc. 

,    weight  x  dist.  weight  from  fulc. 

13.  —  — =dist.  power  from  ful. 

power 

powerxdist.power  from  fulc.      .. 

14. £— =dist. weight  from  fulc. 

weight. 

15.  In  the  first  kind  of  lever,  the  pressure  upon  the  ful- 
crum =  the  sum  of  weight  and  power;  in  the  second  and 
third  =  their  difference. 

16.  If  there  be  several  weights  on  both  sides  of  the  ful- 
crum, they  may  be  reckoned  powers  on  the  one  side  of  the 
fulcrum,  and  weights  on  the  other.     Then,  if  the  sum  of 
the  product  of  all  the  weights  x  their  distances  from  tho 


THE    LEVER.  123 

fulcrum  he  =  to  the  sum  of  the  products  of  all  the  power* 
X  their  distances  from  the  fulcrum,  the  lever  will  be  at  rest, 
if  not,  it  will  turn  round  the  fulcrum  in  the  direction  of  that 
side  whose  products  are  gre;>: 

17.  In  these  calculations,  the  weight  of  the  lever  is  not 
taken  into  account ;  but  if  it  is,  it  is  just  reckoned  like  any 
other  weijrht  or  power  acting  at  the  centre  of  gravity. 

18.  When  two,  three,  or  more  levers  act  upon  each  other 
in  succession,  then  the  entire  mechanical  advantage  which 
they  give,  is  found  by  taking  the  product  of  their  separate 
advantages. 

19.  It  is  to  be  observed,  in  general,  before  applying  these 
observations  to  practice,  that  if  the  lever  be  bent,  the  dis- 
tances from  the  fulcrum  must  be  taken,  as  perpendiculars 
drawn  from  the  lines  of  direction  of  the  weight  and  power 
to  the  fulcrum. 

Ex. — In  a  lever  of  the  first  kind,  the  weight  is  16,  its 
distance  from  the  fulcrum  12,  and  the  power  is  8 ;  there- 
fore, by  No.  13  of  this  chapter, 

— =  24,  the  distance  of  power  from  the  fulcrum. 
8 

In  a  lever  of  the  second  kind,  a  power  of  3  acts  at  a  distance 
of  12 ;  what  weight  can  be  balanced  at  a  distance  of  4 
from  the  fulcrum  ?  Here,  by  No.  12, 

3  x  12        o        •  w  ' 
— =  9,  weight. 

In  a  lever  of  the  third  kind,  the  weight  is  60,  and  its  dis- 
tance 12,  and  the  power  acts  at  a  distance  of  9  from  the 
fulcrum.;  therefore,  by  No.  11, 

60  X  12  . 
=  80,  the  power  required. 

if 

If  there  be  a  lever  of  the  first  kind,  having  three  weights, 
7,  8,  and  9,  at  the  respective  distances  of  6,  15,  and  29, 
from  the  fulcrum  on  one  side,  and  a  power  of  17  at  the  dis- 
tance of  9  on  the  other  side  of  the  fulcrum  ;  then  a  power  is 
,to  be  applied  at  the  distance  of  12  from  the  fulcrum,  on  the 
last  mentioned  side  :  what  must  that  power  be  to  keep  the 
l^ver  in  balance  ? 

Here  (6  x  7)  +  (15  x  8)  +  (29  x  9)  =  423  =  the 
effect  of  the  three  weights  on  the  one  side  of  the  ful- 
crum ;  and  17  X  9  =  153  =  the  effect  of  the  power  on  the 
other  side.  Now,  it  is  clear  that  the  effect  of  the  weight  is 


124 


MECHANICS. 


far  grei  ter  than  the  effect  of  the  power ;  and  the  difference 
423  —  1  53  =  270  requires  to  be  balanced  by  a  power  ap- 
plied at  the  distance  of  12,  which  will  evidently  be  found 
by  dividing  270  by  12,  which  gives  22'5,  the  weight  re- 
quired. 

20.  The  Roman  steel-yard  is  a  lever  of  the  first  kind,  so 
contrived  that  only  one  movable  weight  is  employed. 

The  common  weighing  balance  is  also  a  lever  of  the  first 
kind.  The  requisites  of  a  good  balance  are :  that  the  points 
of  suspension  of  the  scales  and  the  centre  of  motion,  or  ful- 
crum of  the  beam,  be  all  in  one  straight  line — that  the  arms 
of  the  beam  be  equal  to  each  other  in  every  respect — that 
they  be  as  long  as  possible — that  the  centre  of  gravity  of 
the  beam  be  a  very  little  below  the  centre  of  motion — that 
the  beam  be  balanced  when  the  scales  are  empty,  &c. 
But  we  may  ascertain  the  true  weight  of  any  body  even  by 
a  false  balance,  thus :  weigh  the  body  first  in  one  scale, 
then  in  the  other,  and  multiply  their  weights  together ; 
then  the  square  root  of  this  product  will  be  the  true  weight. 


THE  WHEEL  AND  AXLE. 

21.  THE  wheel  and  axle  is  a  kind  of  lever,  so  contrived 
as  to  have  a  continued  motion  about  its  fulcrum,  or  centre 
of  motion,  where  the  power  acts  at  the  circumference  of  the 
wheel,  whose  radius  may  be  reckoned  one  arm  of  the  lever, 
the  length  of  the  other  arm  being  the  radius  of  the  axle, 
on  which  the  weight  acts.  If  the  power  acts  at  the  end 
of  a  handspike  fixed  in  the  rim  of  the  wheel,  then  this  in- 
creases the  leverage  of  the  power,  by  the  length  of  the 
handspike. 

The  wheel  and  axle  consists  of  a  wheel 
having  a  cylindric  axis  passing  through  its 
centre.  The  power  is  applied  to  the  cir- 
cumference of  the  wheel,  and  the  weight 
to  the  circumference  of  the  axle. 

In  the  wheel  and  axle,  an  equilibrium 
takes  place  when  the  power  multiplied  by 
the  radius  of  the  wheel,  is  equal  to  the 
weight  multiplied  by  the  radius  of  the 
axle  ;  or,  P  :  W  : :  CA.  :  CB. 

For  the  wheel  and  axle  being  nothing  else  but  a  lever 
so  contrived  as  to  have  a  continued  motion  about  its  ful- 


THE    WHEEL    AND    AXLE.  125 

cnim  C,  the  arms  of  which  may  be  represented  by  AC  and 
BC,  therefore,  by  the  property  of  the  lever,  P  :  W  : :  CA 
:CB. 

If  the  power  does  not  act  at  right  angles  to  CB,  but 
obliquely,  draw  CD  perpendicular  to  the  direction  of  the 
power,  then,  by  the  property  of  the  lever,  P  :  W  : :  CA  : 
CD. 

22.  It  will  be  easily  seen,  that  if  two  wheels  fastened 
together  and  turning  round  the  same  centre,  be  so  adjusted, 
that  while  they  turn  round  they  will  coil  on  their  cir- 
cumferences strings,  to  which  weights  are  suspended ; 
one  of  those  wheels  being  larger  than  the  other,  the  larger 
wheel  will  coil  up  a  greater  length  of  the  string  than  the 
smaller  one  will  do  in  the  same  time,  and  this  will  depend 
either  on  the  radii  or  circumferences  of  the  two  wheels. 
The  velocity  of  the  weight  will  be  in  proportion  to  the 
length  of  string  coiled  in  a  given  time ;  therefore,  the  ve- 
locity of  the  weight  will  be  greater  as  the  wheel  is  larger. 
Now,  as  in  the  lever  we  saw  that  a  small  weight  required  a 
great  velocity  to  balance  a  large  weight  with  a  small  velo- 
city, we  may  infer,  that  the  rules  given  for  levers  will  also 
apply  to  the  wheel  and  axle  ;  since  the  velocity  of  any  body 
on  a  lever  depends  upon  its  distance  from  the  fulcrum. 

Ex. — A  weight  of  13  Ibs.  is  to  be  raised  at  a  velocity 
of  14  feet  per  second,  by  a  power  whose  velocity  is  20  feet 
per  second  ;  how  great  must  that  power  be  ? 

13x14 

— -- —  =  9-1,  the  power  required. 

If  the  velocity  of  the  weight,  be  to  that  of  the  power,  as 
14  to  20,  and  the  radius  of  the  axle  on  which  the  weight  is 
coiled  be  7  ;  then, 

20  x  7 

=  10,  radius  of  wheel  on  which  the  power  acts. 

If  a  weight  of  36  Ibs,  is  to  be  raised  by  an  axle  3  inches 
diameter ;  what  must  be  the  power  applied  at  the  end  of  a 
handspike  4  inches  long,  fixed  in  the  rim  of  the  wheel  con- 
nected with  the  axle,  the  wheel  being  6  inches  diameter  ? 
/Here  the  handspike  will  increase  the  distance  of  the 
power  from  the  fulcrum,  and  will  add  to  the  diameter  of  th« 
wheel  twice  its  own  length  ;  therefore,  8  -f-  6  =  14  ; — 
hence,  ]  4  :  3  :  :  36  :  7'77,  the  power  required  to  keep  the 
weight  in  equilibrio. 

11* 


126  MECHANICS. 

23.  Wheels  acting  on  each  other  by  teeth  or  bands,  may 
be  easily  calculated  in  the  same  way.     Thus,  if  a  wheel 
which  has  30  teeth,  drives  another  of  10  teeth,  it  is  evident, 
that  as  the  larger  wheel  has  three  times  as  many  teeth  as 
the  smaller,  the  smaller  wheel  will  be  turned  round  three 
times  for  once  that  the  larger  one  is  turned  round ;  so  that 
the  velocities  of  the  wheels  will  be  inversely  as  their  num- 
ber of  teeth.     In  like  manner,  it  is  clear,  that  if  the  larger 
wheel  drives  the  smaller  not  by  teeth  but  by  a  band,  their 
revolutions  will  be  inversely  as  their  circumferences. 

Ex. — The  number  of  teeth  in  one  wheel  are  160,  and 
in  another  driven  by  it  are  20,  and  the  larger  wheel  makes 
12  revolutions  in  a  minute  ;  how  many  does  the  smaller  one 
make  ? 

20  :  160  :  :  12  :  96  =  the  number  of  turns  which  the 
smaller  wheel  makes  in  a  minute. 

24.  The  larger  wheel  is  usually  called  the  wheel,  driver, 
or  leader,  and  the  smaller  one  is  called  the  pinion,  driven 
wheel,  or  follower. 

25.  Let  us  now  see  what  would  be  the  action  of  two 
wheels  and  a  pinion.     If  the  first  wheel  contains  80  teeth, 
the  pinion  12  teeth,  and  second  wheel  36  teeth.    Place  the 
first  wheel  and  the  pinion  on  the  same  axis,  so  that  they 
move  together,  one  revolution  of  the  one  being  in  the  same 
time  as  a  revolution  of  the  other,  and  the  pinion  drives 
the  second  wheel.     If  the  first  wheel  makes  16  revolutions 
in  a  minute,  the  pinion  will  do  the  same,  and  the  pinion 
drives  the  second  wheel  ;    therefore,  36  :   12  :  :   16  :  5  3 
=  the  velocity  of  the  second  wheel.     Place  these  so,  that 
the  teeth  of  the  first  wheel  act  in  the  teeth  of  the  pinion, 
and  these  again  act  in  the  teeth  of  the  second  wheel.     If 
the  first  wheel  make,  as  before,  16  turns  in  a  minute,  then 
the  pinion  will  make  12  :  80  :  :  16  :  106T8^  =  in  a  minute  ; 
consequently,  the  revolutions  of  the  second  wheel  will  be 
36  :  12  :  :  106TR?  :  35*55  =  turns  of  the  second  wheel  in 
a  minute. 

26.  When  there  are  a  number  of 
wheels  A,  B,  C,  D,  E,  acting  on 
the  respective  pinions  o,  b,  c,  r/,  e, 
as  then  the  effect  of  the  whole  may 
be  found  thus  :  if  the  letters  which 

epresent  the  wheels  and  pinions  be 
understood  to  signify  the  number  of  teeth  of  each, 


THE    WHEEL    A>'P    AXLE.  127 

power  xAxBxCxDxE 

=  weight. 
«x6xcx</xe 

If  the  velocity  oi'  the  first  wheel  be  used  instead  of  the 
power  applied,  then  this  rule  will  give  the  resulting  velo- 
city instead  of  the  weight. 

Ex. — If  the  numbers  of  the  teeth  of  the  wheels  are  9,  6 
9,  10,  12,  and  those  of  the  pinions  6,  6,  6,  6 ;  then  if  the- 
power  applied  be  14  Ibs.,  we  have 

14  x  9  X  6  X  9  X  10  X  12 

=  105  Ibs.,  the  weight. 

6x0x6x6x6 

And,  by  the  remark  under  the  rule,  if  the  first  make  14 
revolutions  in  the  minute,  the  speed  of  the  last  will  be  105 
in  the  same  time. 

The  same  rule  will  apply  to  the  case  where  the  wheels 
act  on  each  other  by  ropes  or  straps,  if  the  circumferences 
of  the  wheels  and  pinions  are  used  for  the  number  of  teeth. 

27.  It  often  happens,  in  the  construction  of  machinery, 
that  two  shafts  must  be  connected  by  means  of  toothed 
wheels,  in  such  a  way,  that  the  one  shaft's  velocity  shall 
bear  a  certain  proportion  to  that  of  the  other  shaft ;  and  we 
must  determine  the  numbers  of  leeth  in  each  of  the  con- 
necting wheels  and  pinions. 

Take  the  respective  numbers  of  teeth  in  the  pinions  at 
pleasure,  and  multiply  all  these  together,  and  their  product 
again  by  the  number  of  turns  that  the  one  shaft  is  to  make 
for  one  turn  of  the  other  shaft.  Take,  HOAV,  this  product, 
and  find  all  the  numbers  which  will  divide  it  without  a  re- 
mainder, or  divide  its  divisors  without  a  remainder — always 
excepting  the  number  1.  Arrange  all  these  in  one  line,  and 
separate  them  into  parcels  or  bands,  each  containing  as  many 
numbers,  or  factors  (as  they  are  called)  as  you  please  ;  but 
observing,  that  there  must  be  as  many  bands  as  there  are 
wheels  required  ;  then  the  product  of  the  numbers  in  each 
band  will  £five  the  number  of  teeth  in  the  respective  wheels. 
Thus,  if  one  shaft  is  to  turn  720  times  for  another  shaft's 
once,  and  there  be  interposed  4  pinions,  one  of  which  is 
fixed  to  the  end  of  the  one  shaft,  each  pinion  having-  6  teeth 
or  leaves  :  then,  6x6x6xGx  720  ;  all  the  divisors  or 
f/tors  of  which  arc  3.  2.  3,  2,  3.  2,  3,  2,  2,  2,  3,  5.  2,  2,  3  • 
divided  into  4  bands  at  pleasure,  give  the  number  of 
teeth  in  the  wheels.  Thus, 


128  MECHANICS. 

f2x3x5  =30,  f3x3x5  =  45, 

p-tv,  J  2x2x2x3  =  24,  n  j  3X2X2X2X2  =  48, 
Elther  ^2x2x3x3  =  36,  °M  3x3x2  =18, 

1^2x2x3x3  =  36,  ^3x2x2x2        =24. 

The  application  of  what  we  laid  down  may  he  thus  illus- 
trated. In  finding  the  number  of  teeth  in  the  wheels  of  an 
orrery,  we  extract  from  M arm's  Mechanical  Philosophy. 
"  There  is  considerable  difficulty  in  proportioning  the  num- 
ber of  teeth  in  wheels  for  clocks,  orreries,  &c.  the  problem 
indeed  is  indeterminate  ;  we  shall,  however,  give  an  ex- 
ample, that  will  point  out  a  method  by  which  any  ingenious 
mechanic  may  complete  a  piece  of  machinery,  such  as  an 
orrery,  so  as  to  show,  at  all  times,  in  what  part  of  its  orbit 
any  planet  is.  The  following  example  is  for  Mercury; 
this  planet  goes  round  the  sun  in  87d.  23h.;  now,  as  the 
hour  hand  of  a  clock  goes  round  twice  in  24  hours, 
it  will  make  175  }-j  revolutions  in  87d.  23h.  For  the 
fraction  ||-,  take  any  multiple  of  the  denominator  plus 
or  minus  unity,  and  make  it  the  third  term  of  the  propor- 

472 

tion ;    thus    say,  as  12  :  11  : :  515  :  472  nearly  ;    for  — - 

o  1  o 

is  one  unit  less  in  each  than  a  multiple  of  }*-  by  43  =  — 

516 

r  472       90597 

hence  the  revolutions  become  175 = —  — .    Now  the 

515         olo 

only  difficulty  remaining,  is  to  find  proper  factors  or  divi- 
sors that  will  divide  the  numerator  and  denominator 
without  a  remainder,  in  order  to  determine  the  number  of 
teeth  and  leaves  in  the  wheels  and  pinions.  For  the 
numerator,  the  best  method  I  have  found  is  to  make  trial 
of  the  numbers  2  X  5  or  10,  as  often  as  we  can,  and  if  we 
do  not  succeed,  to  try  successively  the  prime  numbers  3, 
7,  11,  13,  17,  19,  23,  29,  31,  37,  41,  43,  47,  &c.  I  find  by 
trial  the  numerator  will  break  into  the  factors  101  X  39  X 
23  =90597,  I  conclude  then  that  these  numbers  101,  39, 
23,  may  be  the  number  of  teeth  in  three  wheels.  I  can 
easily  break  the  denominator  into  the  numbers  103  and  5; 
but  as  103  is  too  large  for  the  teeth  in  a  pinion,  and  being 
a  prime  number,  another  number  must  be  sought  for  that 
will  answer  the  purpose  better.  Again  say,  as  12  :  11  :  • 

1  f*fy  Q 

1825  :  1673.  the  revolutions  now  become  175      —  —^  or 

1S2«) 


THE    WHEEL    AND    AXLE.  129 

321048 

.  Hence  I  find  by  trial  that  the  numerator  (321048) 

1825 

can  be  broken  into  the  factors  91  X  72  X  49  =  321048, 
which  may  be  three  wheels  having  that  number  of  teeth  in 
each.  Again,  the  denominator  of  the  fraction,  or  1825,  is 
capable  of  being  broken  into  the  factors  73  X  5  X  5  -= 
1825.  Now  the  product  of  the  number  of  teeth  in  all  the 
wheels,  divided  by  the  product  of  the  number  of  teeth  in 
all  the  pinions,  will  give  the  revolutions.  For  example, 
32104  -r-  1825  =  175  revolutions,  llh.  Om.  Is.  58  thirds, 
which  does  not  exceed  the  87d.  23h.  (or  175]  |  revolutions) 
by  two  seconds.  The  numbers  last  found  for  the  wheels 
and  pinions,  may  be  transformed  by  multiplication  into 

98X91X72       144X98X91 

more  convenient  numbers,  as — -     = 

73X10X5        73X10X10 

=  175r.  llh.  Om.  Is.  58th.  either  of  which  will  be  a  train 
of  wheel-work  proper  for  such  a  motion,  and  this  train  may 
be  conveniently  attached  to  the  pinion  of  the  hour-wheel 
of  a  clock.  The  reason  for  finding  a  new  fraction,  will 
appear  evident ;  for  if  we  take  the  original  number 
175|i-  =  2}|1,  we  shall  find  it  impossible  to  break  the 
numerator  into  factors  without  leaving  a  fraction,  which  is 
inconsistent  with  wheel-work,  as  nothing  but  whole  num- 
bers will  answer  the  purpose.  It  is  obvious  that  the  higher 
we  take  a  multiple  of  ji  the  nearer  we  approach  to  the  true 
time  of  revolution,  provided  we  can  break  the  numerator 
and  denominator  into  proper  numbers  for  the  teeth  and 
leaves  of  the  wheels  and  pinions.  It  is  necessary  to  ob- 
serve, that  there  must  be  either  three  wheels  and  three 
pinions,  or,  if  the  numbers  when  broken  be  too  large,  if  we 
can  break  them  into  five  wheels  and  five  pinions,  it  will  be 
the  same  thing ;  because  as  the  hands  of  a  clock  go  round 
with  the  sun,  that  motion  would  make  two  wheels  and  two 
pinions  (attached  Co  the  pinion  on  the  hour  wheel)  go  round 
the  contrary  way  to  what  they  ought;  but  three  or  five  will 
answer  the  intended  purpose." 

28.  As  the  subject  of  wheel-work  is  of  the  greatest  im- 
portance to  mechanics,  we  shall  resume  il  in  a  more 
advanced  part  of  this  work,  where  it  maybe  more  properly 
introduced. 


130 


MECHANICS. 


THE   PULLEY. 

29.  IF  a  rope  or  string  pass  round  the  groove  or  rim  of  a 
wheel,  movable  round  an  axle,  with  a  power  ?i  the  one  end 
of  the  string  or  rope,  and  a  weight  at  the  other, — such  a 
machine  is  called  a  Pulley.  The  axis  of  the  pulley  may  be 
either  fixed  or  movable.  If  the  axis  of  the  pulley  be 
fixed,  it  only  serves  to  change  the  direction  of  the  power's 
action ;  but  if  it  be  movable,  the  power  acts  with  an  ad- 
vantage of  two  to  one. 

The  accompanying  engraving  exhibits  various  forms  of 
the  pulley.  AB  is  a  beam  from  which  they  are  suspended. 


No.  1,  is  the  fixed  pulley  in  which  there  is  no  other  ad- 
vantage gained  than  that  the  power.  P  and  weight  W  move 
in  a  contrary  direction.  No.  2,  is  a  movable  pulley,  in 
which  the  power  P  by  moving  upwards  raises  the  pulley, 
to  the  block  of  which  the  weight  W  is  attached  ;  but  the 
one  end  of  the  string  being  attached  to  the  beam  AB,  the 
power  must  move  twice  as  fast  as  the  weight,  and  there 
will  be  a  gain  of  power  proportional.  No.  3,  is  a  combi- 
nation of  two  movable  pulleys,  in  which  the  gain  of  power 
will  be  four ;  and  No.  4  is  a  combination  of  two  fixed  and 
two  movable  pulleys,  in  which  the  gain  of  power  will  be 
the  same  as  in  No.  3. 

30.  If  in  a  system  of  pulleys,  where  each  pulley  is  em- 
braced by  a  cord,  attached  at  one  end  to  a  fixed  point,  and 
at  the  other  to  the  centre  of  the  movable  pulley  next  above 
it,  and  the  weight  is  hung  to  the  lowest  pulley ;  then  the 
effect  of  the  whole  will  be  =  the  number  2  multiplied  by 
itself,  as  .many  times  as  there  are  movable  pulleys  in  the 
system :  thus,  if  there  be  4  movable  pulleys,  then  2x2 


THE    PULLEY.  131 

<  2  X2  =10:  wherefore,  if  the  weight  be  one  lb.,  it  will 
oe  sustfined  by  a  power  of  one  oz.  avoirdupois. 

31.  When  there  are  any  number  of  movable  pulleys  on 
one  block,  and  as  many  on  a  fixed  block,  the  pulleys  are 
called  Sheeves,  and  the  system  is  called  a  Muffle  ;  and  the 
weight  is  to  the  power  inversely  as  one  is  to  twice  the 
number  of  movable  pulleys  in  the  system,  or 

the  weight  to  be  raised 
twice  the  number  of  mov.  pulleys  ~ 
Ex. — In  a  muffle  where  each  block  has  4  sheeves,  one 
block  being  h'xed  and  the  other  movable,  a  weight  of  112 
Ibs.  is  to  be  raised  ;  how  great  must  be  the  power  ? 

112 

-p — =  14  Ibs.,  the  power  required. 

If  a  power  of  236  Ibs.  is  to  be  applied  to  a  tackle  con 
nected  with  two  blocks  of  pulleys,  otic  fixed,  consisting  of 
6,  and  another  movable,  of  5  pulleys  ;  what  weight  can  be 
raised  ? — (Here  the  rule  above  must  be  reversed.) 

Therefore  236  X  10  =  2360  Ihs.,  the  weight. 

REMARK. — In  all  the  above  cases  of  the  pulley,  the  strings, 
cords,  or  ropes,  are  supposed  to  act  parallel  to  each  other ; 
when  this  is  not  the  case,  the  relation  of  power  and  weight 
may  be  found  by  applying  the  principle  of  the  parallelo- 
gram of  forces ;  thus,  draw  ab  in  tbe  direction  of  the 
power's  action  and  of  that  length,  taken  from 
a  scare  of  equal  parts,  which  expresses  the 
quantity  of  that  power ;  next,  draw  bd  a  per- 
pendicular to  the  horizon,  and  from  «  draw 
ad  parallel  to  be,  the  direction  of  the  string, 
which  is  fastened  at  c:  then  the  power  is  to 
the  weight,  as  ba  is  to  bd;  and  the  strain  on  the  hook  at  c, 
is  as  ad  to  db, — these  lines  being  all  measured  on  the  same 
scale  of  equal  parts. 

It  may  be  further  observed,  that  the  pulley  is  a  species 
of  lever  of  the  second  kind  ;  where  the  point  at  which  the 
string  is  fastened  may  be  called  the  fulcrum  ;  the  axis  of 
the  pulley  the  place  of  the  weight,  and  the  place  of  the 
power  the  other  end  of  the  lever ;  or,  the  diameter  of  the 
puiley  may  be  reckoned  the  length  of  the  leverf  the  weight 
being  in  the  middle. 


132  MECHANICS 

THE  INCLINED  PLANE. 

32.  WHEN  a  power  acts  on  a  body, 
on  an  inclined  plane,  so  as  to  keep  that  C 
body   at    rest ;    then    the    weight,    the 
power,  and   the  pressure  on  the  plane, 
will  be  as   the  length,  the    height,  and 

the  base  of  the  plane,  when  the  power  acts  parallel  to  tht 

plane ;  that  is, 

The  weight  f  "]  AC, 

The  power  -<  will  be  as  >BC, 

The  pressure  on  the  plane  (_  J  AB. 

These  properties  give  rise  to  the  following  rules  : — 

weight  X  height  of  plane 

power  = ,     — 

length  01  plane 

power  x  length  of  plane 

weight  =  —    : — — , jr-i —  — 

height  of  plane 

weight  X  base  of  plane 

pressure  on  the  plane  = • 

length  of  plane 

33.  The  force  with  which  a  body  endeavours  to  descend 
down  an  inclined  plane,  is  as  the  height  of  the  plane. 

When  the  power  does  not  act  parallel 
to  the  plane,  then  from  the  angle  C  of 
the  plane,  draw  a  line  perpendicular 

to  the  direction  of  the  power's  action  ; 

then   the    weight,  the    power,   and   the     B  * 

pressure  on  the  plane,  will  be  as  AC,  CB,  AB. 

When  the  line  of  direction  of  the  power  is  parallel  to  the 
plane,  the  power  is  least. 

34.  If  two  bodies,  on  two  Inclined  planes,  sustain  each 
other,  by  means  of  a  string  over  a  pulley,  their  weights 
will  be  inversely  as*  the  lengths  of  the  planes. 

35.  In  the  exercises  on  inclined  planes,  it  is  often  neces- 
sary to  find  the  length  of  the  base,  and  height,  or  length  of 
the  plane.     Any  two  of  these  being  given,  the  third  may 
be  found — and  this  is  done  on  the  principle  stated  in  Geo- 
metr)r,  that  the  hypotenuse  3  of  a  right-angled  triangle  (the 
length  of  the  plane)  is  equal  to  the  base  a  +  height3. 

Ex. — The  height  of  an  inclined  plane  is  20  feet,  and  its 
length  TOO  ;  what  is  the  pressure  on  the  plane  of  a  weight 
of  1000  Ibs.? — Here  we  must  first  ascertain  the  base, 
(100a— 202)  k  =  97-98  =  the  base  of  the  plane  ;  and  from 


THE    INCLINED    PLANE.  133 

what  has  been  said  above,  100  :  1000  ::  97'98  :  979-8  the 
pressure  upon  the  plane;  also  100  :  20  : :  1000  •  200,  the 
power  necessary  to  keep  the  body  from  rolling  down  the 
plane. 

If  a  wagon  of  3  cwt.  on  an  inclined  railway  of  10  feet  to 
the  100,  be  sustained  by  another  on  an  opposite  railway  of 
10  feet  to  90  of  an  incline ;  what  is  the  weight  of  the  second 
wagon  ? — Here  100  :  90  :  :  3  :  2-7  cwt.  =  the  weight  oi 
the  second  wagon. 

36.  The  space  which  a  body  describes  upon  an  inclined 
plane,  when  descending  on  the  plane  by  the  force  of 
gravity,  is  to  the  space  which  it  would  fall  freely  in  the 
same  time,  as  the  height  is  to  the  length  of  the  plane ;  and 
the  spaces  being  the  same,  the  times  will  be  inversely  in 
this  proportion. 

Ex. — If  a  body  roll  down  an  inclined  plane  320  feet  long, 
and  26  feet  in  height;  what  space  will  it  pass  down  the 
plane  in  one  second,  by  the  force  of  gravity  alone  ? 
320  :  26  : :  16  :  1-3  foot  =  the  answer. 

This  subject,  as  connected  with  railways,  will  be  resumed 
when  we  come  to  treat  of  friction  and  railways. 


THE    WEDGE. 

37.  THE  wedge  is  a  triangular  prism,  formed  either  of 
wood  or  metal,  whose  great  use  is  to  split  or  raise  timber, 
stones,  &c. 

The  circumstances  in  which  it  is  applied  are  such  that 
it  is  not  easy  to  devise  a  general  rule  to  determine  the 
amount  of  its  action.  The  wedge  has  a  great  advantage 
over  all  the  other  mechanical  powers,  in  consequence  of  the 
way  in  which  the  power  is  applied  to  it,  namely,  by  per 
cussion,  or  a  stroke,  so  that  by  the  blow  of  a  hammer, 
almost  any  constant  pressure  may  be  overcome. 


THE    SCREW. 

38.  THE  screw  is  a  kind  of  continued  inclined  plane, 
being  an  inclined  plane  rolled  about  a  cylinder — the 
height  of  the  plane  being  the  distance  between  the  centres 
of  two  threads,  and  its  length  the  circumference ;  hence, 

12 


134  MECHANICAL    CENTRES 

the  rule  to  find  the  power  of  a  screw  pressing  either  up- 
wards or  downwards,  is  as  the  distance  between  two  threads 
of  the  screw  is  to  the  circumference  where  the  power  i? 
applied  :  thus.,  if  the  distance  of  the  centres  of  two  threads 
of  the  screw  be  |  of  an  inch,  and  the  radius  of  the  hand- 
spike attached  to  the  screw  be  24  inches  ;  the  circumference 
j>(  the  screw  will  be  150J-  inches,  nearly:  therefore, 
|  :  150|  :  :  1  :  603}  ;  and  if  the  power  applied  be  150 
Ibs.,  the  force  of  the  screw  will  therefore  be  603£  X  150 
==  90480  Ibs. 

39.  REMARKS  ON  THE  MECHANICAL  POWERS. — The  me- 
chanical powers  may  be  variously  modified  and  applied, 
but  still  they  form  the  elements  of  all  machinery.  In  our 
calculations  of  their  effects,  we  have  not  made  allowance 
for  friction,  or  the  resistance  arising  from  one  body  rubbing 
against  another — a  subject  which  will  be  discussed  hereafter. 
The  justice  of  the  remark  made  before,  will  now  be  seen 
to  hold  generally,  that  of  the  two — velocity  and  power — > 
whatever  we  gain  in  the  one,  we  lose  in  the  other ;  or,  as 
power  and  weight  are  opposed  to  each  other,  there  will 
always  be  an  equilibrium  between  them,  when  the  power 
X  its  velocity  =  the  weight  x  its  velocity,  that  is,  when 
the  momentum  of  the  one  is  equal  to  the  momentum  of  the 
other. 

All  the  advantage  that  we  can  obtain  from  the  mechanical 
powers,  or  their  combinations,  is  to  raise  great  weights,  or 
overcome  great  resistances,  and  this  must  be  done  at  the 
expense  of  time;  or,  to  generate  rapid  velocities,  as  in 
turning-lathes,  or  cotton-spinning  machinery,  and  this  is 
done  at  the  expense  of  power. 


MECHANICAL  CENTRES. 

1.  THESE  are  the  centres  of  gravity,  oscillation,  percus- 
sion, and  gyration. 

THE  CENTRE  OF  GRAVITY. 

2.  THERE  is  a  certain  point  in  every  body,  or  system  of 
bodies  connected  together ;  .which  point,  if  suspended,  the 


THE  CENTRE  OF  GRAVITY.          135 

body  or  system  of  bodies  will  remain  at  rest  when  acted 
upon  by  the  force  of  gravity  alone  ; — this  point  is  called 
the  Centre  of  Gravity.  If  a  body  or  system  of  bodies  be 
suspended  by  any  other  point  than  the  centre  of  gravity, 
such  body  or  system  of  bodies  will  move  round  that  point, 
until  the  centre  of  gravity  be  in  a  vertical  line  with  the 
point  of  suspension.  If  a  body  be  sustained  from  falling 
by  two  forces,  the  lines  of  direction  in  which  these  two 
forces  act,  will  meet  in  the  centre  of  gravity  of  the  body, 
or,  in  the  vertical  line  which  passes  through  it. 

3.  It  is  often  useful  in  calculation  to  consider  the  whole 
weight  of  a  body  as  placed  in  its  centre  of  gravity,  but  it 
is    to  be   remembered,    that   gravity   and    weight   do    not 
signify  the  same  thing — gravity  is  the  force  by  means  of 
which   bodies,  if  left  to   themselves,  fall   to  the  earth  in 
directions  perpendicular  to  the  earth's  surface  ;  weight,  on 
the  other  hand,  is  the  resistance  or  force  which  must  be 
exerted,  to  prevent  a  given  body  from  obeying  the  law  of 
gravity. 

4.  To  find  the  centre  of  gravity  of  any  plane  figure,  me- 
chanically :   Suspend  the  figure  by  any  point  near  its  edge, 
and  mark  the  direction   of  a  plumb-line  hung  from  that 
point,  then  suspend   it  from  some  other  point,  and  mark 
the   direction   of    the    plumb-line    in   like    manner.     The 
centre  of  gravity  of  the  figure  will  be  in  that  point  where 
the  marks  of  the  plumb-line  cross  each  other.    For  instance, 
if  we  wish  to  find  the  centre  of  gravity  of  the  arch  of  a 
bridge,  we  draw  the  plan  upon  paper  to  a  certain  scale, 
cut  out  the  figure,  and  proceed  with  it  as  above  directed ; 
and  by  means  of  the  plumb-line  from  the  points  of  sus- 
pension, its  centre  of  gravity  will  be  found  ;  whence,  by 
measuring  the  relative  position  of  this  centre  in  the  plan 
by  the  scale,  we  may  determine  by  comparison  its  position 
in  the  structure  itself. 

5.  We  can  find  the  centre  of  gravity  of  many  figures  by 
calculation. 

6.  The  centre  of  gravity  of  a  line,  parallelogram,  prism, 
cylinder,   circle,  circumference   of  a    circle,  sphere,    and 
regular  polygon,  is  the  geometrical  centre  of  these  figures 
respectively. 

ff.  To  find  the  centre  of  gravity  of  a  triangle — draw  a  line 
from  any  angle  to  the  middle  of  the  opposite  side,  then  f 


13t  MECHANICAL    CENTRES. 

of  this  line  from  the  angle  will  be  the  position  of  the  centre 
of  gravity. 

8.  For  a  trapezium, — draw  the  two  diagonals,  and  find 
the  centres  of  gravity  of  each  of  the  four  triangles  thus 
formed,  then  join  each  opposite  pair  of  these  centres  of 
gravity,  and  the  two  joining  lines  will  cut  each  other  in 
the  centre  of  gravity  of  the  figure. 

9.  For  the  cone  and  pyramid, — the  centre  of  gravity  is  in 
the  axis,  at  the  distance  of  ?  of  the  axis  from  the  vertex 

10.  For  the  arc  of  a  circle, — 

radius  of  circle  X  chord  of  arc  -   , 

— . =   distance  of  the 

length  ot  arc 

centre  of  gravity  from  the  centre  of  the  circle. 

11.  For  the  sector  of  a  circle, — 

2  X  chord   of  arc  x  radius  of  circle 

— : — ? =  distance  of 

3  x  length  of  arc 

the  centre  of  gravity  from  the  centre  of  the  circle. 

12.  For  a  parabolic  space, — the  distance  of  the  centre  of 
gravity  from  the  vertex  is  £  of  the  axis. 

13.  For  a  paraboloid, — the  centre  of  gravity  is  f  of  the 
axis  from  the  vertex. 

14.  For  two  bodies, — if  at  each  end  of  a  bar  a  weight  be 
hung,  the  common  centre  of  gravity  will  be  in  that  point 
which  divides  the  bar,  in  the  same  ratio  that  the  weights 
of  the  bodies  bear  to  each  other,  and  this  point  will  be 
nearest  the  heavier  body. 

Examples. — If  the  line  drawn  from  the  middle  of  the  base 
of  a  triangle  to  the  opposite  angle  be   15,  then  we  have 

15 

—  X  2  =  10  =  the  distance  of  the  centre  of  gravity  from 
o 

the  vertical  angle. 

If  the  height  of  a  cone  be  24  inches,  then  we  have 

24 

—  X  3  =  18  =  the  distance  of  the  centre  of  gravity  from 

the  vertex. 

If  the  length  of  the  arc  of  a  circle  be  157-07,  and  the 
chord  153-07,  and  radius  200  ;  then, 

200  X  153-07 
— 1  ^7-07 ==  "***"  =  distance  of  the  centre  of 

gravity  from  the  centre  of  the  circle. 

If  there  be  the  sector  of  a  circle  of  which  the  chord 


OSCJU.ATiON    AND    I'M  ISC  1VSJON.  137 

radius,  and  length  of  :irc,  arc  the  same  as  in  the  last  ex 
anipli-,  we  have 

2  X  153-07  X  200        lonQ 

3  X  l-)?-07       ~  =  =  distance   of   the 

centre  of  gravity  from  the  centre  of  the  circle. 

In  a  parabolic  space,  if  the  axis  be  25  inches  long,  then 

25 

—  X  3  =  15  =  the  distance  of  the  centre  of  gravity  from 
o 

the  centre. 

30 

In  a  parabdoid,  if  the  axis  be  30,  then  we  have  —  X  2 

3 

=  20  =  the  distance  of  the  centre  of  gravity  from  the  vertex. 

A  bar  of  wood,  24  feet  long,  has  a  weight  suspended  at 
each  end,  that  at  one  end  being  16  Ibs.,  and  the  other  4 : 
then,  we  have  20  :  24  :  :  16  :  19-2 

and  20  :  24  : :     4  :     4-8 

the  distances  of  the  weights  from  the  common  centre  of 
gravity,  the  greater  weight  being  least  distant.  Hence  we 
see,  that  19-2  -f  4-8  =  24,  the  whole  length  of  the  bar; 
and  also  4x19-2  =  16x4-8  =  76-8  ;  so  that  the  prin- 
ciple of  virtual  velocities,  stated  before,  holds  good  here  also ; 
and  here  it  may  be  observed,  that  it  is  of  the  greatest  im- 
portance to  trace  any  leading  principle  of  this  kind  through 
its  various  applications,  as  it  serves  to  link  together  and 
harmonize  the  whole,  and  enables  us  to  apply  and  remember 
it  with  greater  facility. 

It  is  often  necessary  to  determine  the  centre  of  gravity 
experimentally,  as  in  many  cases  it  cannot  be  conveniently 
done  by  calculation.  To  maintain  the  firmness  of  any 
body  resting  on  a  base,  it  is  necessary  that  the  perpendicu- 
lar drawn  from  the  centre  of  gravity  of  the  body,  to  the 
base  on  which  it  rests,  be  within  that  base  ;  and  the  body 
will  be  the  more  difficult  to  overset,  the  nearer  that  per- 
pendicular is  to  the  centre  of  the  base,  and  the  more  ex- 
tensive the  base  is,  compared  to  the  height  of  the  centre  of 
gravity. 


T* HE  CENTRE  OF  OSCILLATION.— THE  PENDULUM,  AND 
CENTRE  OF  PERCUSSION. 

1.  THE  centre  of  oscillation  in  a  vibrating  body,  is  that 
point  in  the  axis  of  vibration,  in  which,  if  the  whole  matter 
12* 


138  MECHANICAL    CENTRES. 

contained  in  the  body  were  collected,  and  acted  upon  by 
the  same  force,  it  would,  if  attached  to  the  same  axis  of 
motion,  perform  its  vibrations  in  the  same  time.  The 
centre  of  oscillation  is  always  situated  in  the  straight  line 
which  passes  through  the  centre  of  gravity,  and  is  perpen- 
dicular to  the  axis  of  motion.  It  will  be  seen  by  these 
remarks,  that  the  subject  of  pendulums  must  be  considered 
here. 

2.  In  theory,  a  simple  pendulum  is  a  single  weight,  con- 
sidered as  a  point,   hanging  at  the  lower  extremity  of  an 
inflexible  right  line,  having  no  weight,  and  suspended  from 
a  fixed  point  or  centre,  about  which  it  vibrates,  or  oscil- 
lates ;  a  compound  pendulum,  on  the  other  hand,  consists 
of  several  weights,  so  connected  with  the  centre  of  suspen- 
sion, or  motion,  as  to  retain  always  the  same  distance  from 
it,  and  from  each  other. 

3.  If  the  pendulum  be  inverted,  so  that  the  centre  of 
oscillation  shall  become  the  centre  of  suspension,  then  the 
former  centre  of  suspension  will  become  the  centre  of  os- 
cillation, and  the  pendulum  will  vibrate  in  the  same  time : 
this  is  called  the  reciprocity  of  the  pendulum  ;  and  it  is  a 
fact  of  the  greatest  utility,  in  experimenting  on  the  lengths 
of  pendulums. 

4.  Of  the  simple  pendulum  we  may  observe,  that  its 
length,  when  vibrating  seconds,  must  in  the  first  place  be 
determined  by  experiment,  as  it  vibrates  by  the  action  of 
gravity, — which  force  differs  at  different  distances  from  the 
pole  of  the  earth.     By  the  latest  experiments,  the  length  of 
the  seconds'  pendulum  in  the  latitude  of  London,  has  been 
found  to  be  39-1393  inches,  or  3-2616  feet;  the  length  at 
the  equator  is  nearly  39-027,  and  at  the  pole  39-197  inches. 
The  length  for  the  latitude  of  London  may  be  taken  for  all 
places  in  Britain,  without  any  material  error. 

5.  The  times  of  vibration  of  two  pendulums,  are  directly 
proportional  to  the  square  roots  of  the  lengths  of  these  pen- 
dulums. 

6.  Thus  :    what  will  be  the  time  of  one  vibration  of  a 
pendulum  of  12  inches  long  at  London? 

x/39-1393  :  x/  12  ::  I  :  0-5537  =  time  of  one  vibration. 

If  the  pendulum  be  36  inches  long, 

v/39-1393  :  v/  36  ::  1  :  0-9599  =  time  of  one  vibration. 

7.  The  lengths  of  the  pendulums  are  to  each  other  in- 


OSCILLATION,  AND    PERCUSSION.  139 

trersely  as  the  squares  of  the  numbers  of  vibrations  made  in 
a  given  time. 

What  is  the  length  of  a  pendulum  vibrating  half-seconds 
or  making  30  vibrations  in  a  minute  ? 

(60)a  :  (30)a  :<  39-1393  :  9-7848  =  length  in  inches. 
The  length  of  a  pendulum  to  make  any  given  number  of 
vibrations  in  a  minute,  may  be  easily  found  by  the  following 
short  rule  :  — 

140850 

i-   -i  —  "•  --  «  =  length. 
number  of  vibrations3 

Thus  a  pendulum  to  make  50  vibrations  in  a  minute,  will 
be 

140850       140850        _ 


SO*  2500 

8.  All  the  rules  for  simple  pendulums  may  be  expressed 
as  follows  : 

The  time  of  one  vibration  in  seconds  of  any  pendulum  is 
_  1  _ 

number  of  vibrations  in  one  second 

I 

or 


|/ the  length  of  the  pendulum\ 
\>  39-1393 


Exam.  —  If  the  number  of  vibrations  of  a  pendulum  be 
•6256,  then 

-—  —  —=  1-598  =  the  time  of  one  vibration. 
"o25o 

Or,  if  the  length  of  the  pendulum  be  100  inches,  then 


The  length  of  a  pendulum  in  inches  is 

=  39-1393  x  time  of  one  vibration8; 

39-1393 

or  -      --  • 
number  of  vibrations' 

Exam.  —  If  the  time  of  one  vibration  be  1'598  ;  find  the 
length.         39-1393  X  l'598a  =  100,  length  of  pend. 

Or.  if  the  number  of  vibrations  in  a  second  be  as  above, 
•6856,  then  we  have  — 
39-1393 
.6256a    =  100,  length  of  pendulum. 


14C  MECHANICAL    CENTRES. 

The  number  of  vibrations  in  a  second  may  be  found  thus : 

|          39-1393 

.    . r — f j- : —  =  number  ol  vibrations  ; 

\length  of  pendulum 

or,  the  number  of  vibrations  in  a  second  is 


time  of  one  vibration          • 
Tf  the  time  of  one  vibration  be,  as  above,  1*598 ;  then 

-  =  '6256,  number  of  vibrations  ; 
i *oy o  ' 

or,  if  the  length  of  100,  we  have" 

~)  =  —  • 

When  a  clock  goes  too  fast  or  too  slow,  so  that  it  shalj 
lose  or  gain  in  twenty-four  hours,  it  is  desirable  to  regulate 
the  length  of  the  pendulum  so  that  it  shall  go  right.  The 
pendulum  bob  is  made  capable  of  being  moved  up  or  down 
on  the  rod  by  means  of  the  screw.  If  the  clock  goes  too 
fast,  the  bob  must  be  lowered,  and  if  too  slow,  it  must  be 
raised ;  and  we  have  this  rule  :  number  of  threads  in  an 
inch  of  the  screw  X  the  time  in  minutes  that  the  clock 
loses  or  gains  in  24  hours  ;  this  product  divided  by  37  will 
give  the  number  of  threads  that  the  bob  must  be  screwed 
up  or  down,  so  that  the  clock  shall  go  right. 

Ex. — If  the  rod  have  a  screw  70  threads  in  the  inch,  and 
the  pendulum  is  too  long,  so  that  the  clock  is  12  minutes 
slow  in  24  hours  ;  then  we  have 

2  x  70  x  12 

— =  45 Jf  =  threads  we  must  raise  the  bob, 

o7 

so  that  the  clock  shall  go  right. 

9.  It  is  often  desirable  that  a  pendulum  should  vibrato- 
seconds,  and  yet  be  much  shorter  than  39-1393  inches; 
Avhich  may  be  done  by  placing  one  bob  on  the  rod  above 
the  centre  of  suspension,  and  another  below  it :  then,  having 
the  distances  of  the  weights  from  the  centre  of  suspension, 
we  may  find  the  ratio  which  the  weights  should  bear  to 
each  other  by  this  rule.  Call  D  the  distance  of  the  lower, 
and  d  the  distance  of  the  upper  weight,  from  the  centre  of 
suspension ;  then, 

39-1393  x  D  — Ds 


OSCILLATION    ANU    PERCUSSION.  141 

a  number  which,  when  multiplied  by  the  lower  weight, 
will  give  the  hi-licr.  1)  anil  d  are.  taken  in  inches. 

Ex. — In  a  pendulum  having  two  bobs,  the  one  12  inches 
oelow  the  centre  of  suspension,  and  the  other  9-6  inches 
above  the  same  centre,  the  lower  weight  being  8  ounces ; 
what  is  the  upper  weight? 

39-1393  X  12  — 123     =  . 

39-1393  x  9-6  +  9-tt* 

then,  0-696  X  8  =  5-568  ounces  =  the  weight  of  the 
upper  bob. 

•  10.  If  a  common  walking-stick  be  held  in  ^ie  hand,  and 
struck  against  a  stone,  at  different  points  of  its  length,  it 
will  be  found  that  the  hand  receives  a  shock  when  it  is 
struck  at  any  part  of  the  stick,  but  at  one  particular  point, 
at  which,  if  the  stick  be  struck,  the  hand  will  receive  no 
shock.  This  point  is  called  the  centre  of  Percussion,  and  is 
usually  defined  thus  : — The  centre  of  percussion  is  that  point 
in  a  body  revolving  about  an  axis,  at  which,  if  it  struck  an 
immovable  obstacle,  all  the  motion  of  the  body  would  be 
destroyed,  so  that  it  would  incline  neither  way  after  the 
stroke. 

11.  The  distance  of  the  centre  of  percussion  from  the 
axis  of  motion,  is  the  same  as  the  distance  of  the  centre  of 
oscillation  from  the  centre  of  suspension ;  and  the  same 
rules  serve  for  both  centres. — See  Oscillation. 

12.  The  distance  of  either  of  these  centres  from  the  axis 
of  motion,  is  found  thus  : — 

13.  If  the  axis  of  motion  be  in  the  vertex  of  the  figure, 
and  the  motion  be  flatwise ;  then, 

14.  In  a  right  line,  it  is  =  *  of  its  length  ; 

In  an  isosceles  triangle  =  |  of  its  height ; 
In  a  circle  =  f  of  its  radius  ; 
In  a  parabola  =  i  of  its  height. 

15.  But  if  the  bodies  move  side  wise,  we  have  it 
In  a  circle  =  |  of  the  diameter ; 

In  a  rectangle  suspended  by  one  angle  =  f  of  the 
diagonal. 

16.  In  a  parabola  suspended  by  its  vertex, 

=  4  axis  +  i  parameter;* 
hot  if  suspended  by  the  middle  of  its  base, 
=  ^  axis  -f  5  parameter. 


142  MECHANICAL    CENTRES. 

3  X  arc  X  radius 


17.  In  the  sector  of  a  circle  = 

18.  In  a  cone  =  f  axis  -f 


4  x  chord 
(radius  of  base)' 
5  x  axis 


10   T  u  2  x  ,•  T       , 

19.  In  a  sphere  =  — r — r  +  radius  -}-  </,  where 

5  (a  -f  radius) 

e?  is  the  length  of  the  thread  by  which  it  is  suspended. 

20.  We  have  given  these  rules  for  the  sake  of  reference 
but  we  shall  illustrate  by  examples  the  most  useful. 

Examples. — What  must  be  the  length  of  a  rod  without  a 
weight,  so  mat  when  hung  by  one  end  it  shall  vibrate 
seconds  ? 

To  vibrate  seconds,  the  centre  of  oscillation  must  be 
39-1393  inches  from  that  of  suspension;  hence,  as  this 
must  be  f  of  the  rod,  2:3::  39-1393  :  58-7089  inches  = 
the  length  of  the  rod. 

What  is  the  centre  of  percussion  of  a  rod  46  inches  long? 
^  X  46  =  30|  inches  from  the  axis  of  motion. 

In  an  isosceles  triangle,  suspended  by  one  angle,  and 
oscillating  flatwise,  the  height  is  24  feet ;  what  is  the  dis- 
tance of  the  centre  of  percussion  from  the  axis  of  motion  ? 
|  X  24  =  18  feet. 

In  a  sphere  the  diameter  is  14,  and  the  string  by  which 
the  sphere  is  suspended  is  20  inches ;  therefore, 

2  v  7fl  98 

7  +  20=— -  +  27  =  27-725  ; 


5  (20  +  7)    '  135 

so  that  the  centre  of  oscillation  or  percussion  is  farther  from 
the  axis  of  motion  than  the  centre  of  the  sphere,  by  7'725 
inches. 


THE  CENTRE  OF  GYRATION  AND  ROTATION. 

21.  IT  will  be  seen,  that  the  last  two  centres  refer  to 
bodies  in  motion  round  a  fixed  axis,  and  belonging  to  the 
same  class :  there  is  yet  another  centre  to  be  considered, 
of  the  utmost  impof  tance  to  the  practical  mechanic.  We 
saw,  in  determining  the  centre  of  oscillation,  that  we  were 
finding  a  point  in  which,  if  all  the  matter  of  the  body  were 
collected,  the  motion  would  be  the  same  as  that  of  the  body 
—which  motion  was  caused  by  the  action  of  gravity ;  but 


GYRATION  AND  ROTATION.  143 

when  the  body  is  put  in  motion  by  some  other  force  than 
gravity,  the  point  in  question  becomes  the  centre  of  Gyra- 
tion. The  centre  of  ny ration  may  therefore  be  defined, 
that  point  in  a  body  or  system  of  bodies  revolving  round 
an  axis,  in  which  point,  if  all  the  matter  in  the  body  or 
system  of  bodies  were  collected,  the  same  number  of  revo- 
lutions in  a  given  time  would  be  generated  by  the  applica- 
tion of  a  given  force,  as  would  be  generated  by  the  same 
force  applied  to  the  body  or  system  of  bodies  itself. 

22.  The  position  of  the  centre  of  gyration  is  a  me?wi  pro- 
portional between  the  centres  of  oscillation  and  gravity. 

23.  The  centre  of  gyration  of  the  following  bodies  may 
be  found  by  these  rules  : — 

24.  For  a  straight  line  or  cylinder,  whose  axis  of  motion 
is  in  one  end,  =  length  x  0*5775. 

25.  For  a  cylinder  or  plane  of  a  circle,  revolving  about 
the  axis,  or  the  circumference  about  the  diameter,  =  radius 
X  0-7071. 

26.  For  the  plane  of  a  circle  about  its  diameter  =  5 
'radius. 

27.  For  the  surface  of  a  sphere  about  its  diameter  =a 
radius  X  -8165. 

28.  For  a  solid  sphere  or  globe,  about  its  diameter  = 
radius  X  -6324. 

29.  For  the  circumference  of  a  circle  upon  a  perpendicu- 
lar axis  passing  through  the  centre  =  radius. 

Ex. — What  is  the  distance  of  the  centre  of  gyration 
from  the  centre  of  motion,  of  a  rod  58*7089  inches  long? 
Here  58-7089  X  -5775  =  33-9044. 

In  a  wheel  of  uniform  thickness,  revolving  about  its  axis, 
the  diameter  is  36  inches ;  hence  18  X  '7071  =  12  =  dis- 
tance of  the  centre  of  gyration  from  the  axis. 

In  a  solid  globe  revolving  about  its  diameter,  which  is 
2  feet,  the  distance  of  the  centre  of  gyration  is  =  12  X 
•6324  =  7-5888  inches. 

30.  Effects  are  proportional  to  their  causes  ;  the  motion 
'generated   in  any  body  is  proportional  to  the  force  which 
.produces  that  motion  ;  hence  we  see,  that  all  constant  forces 
may  be  compared  to  the  force  of  gravity.     And  it  is  often 
u/eful  to  know  the  time  in  which  a  revolving  body  of  a 
certain  weight,  acted  upon  by  a  known  constant  force,  will 
acquire   a  given   velocity.     The  principles  we  have  laid 


.44  MECHANICAL    CENTRES. 


in  discussing  the  inclined  plane,  will  here  be  found 
serviceable. 

As  the  weight  of  the  body  moved, 

fs  to  the  weight  or  force  causing  it  to  move, 

So  is  the  length  of  an  inclined  plane,  such  that  the 

given  force  would  just  support  the  body  upon  it, 
To  the  height  of  the  plane. 

l^ow,  if  in  a  wheel  6  feet  diameter,  whose  weight,  400  Ibs., 
if  turned  by  a  force  of  56  Ibs.,  acting  at  the  distance  of  18 
inches  from  its  centre  of  motion,  its  centre  of  gyration  being 
b  feet  from  the  same  centre  ;  what  will  be  the  time  required 
to  give  by  this  force  a  velocity  of  20  feet  per  second  at  the 
centre  of  gyration.  Here,  by  the  lever, 

18  X  56 

-gg—  =l«ttlbs.- 

the  force  exerted  at  the  centre  of  gyration.  We  now  wish 
to  know  the  length  of  time  in  which  a  body  would  acquire 
a  velocity  of  20  feet  per  second,  on  an  inclined  plane, 
whose  length  is  to  its  height  as  400  is  to  16^f  ;  wherefore, 
by  the  laws  of  falling  bodies,  we  have 
16-8 


the  time  required  to  fall  perpendicularly  ;  therefore,  by  the 
.nclined  plane,  we  have,  20  :  400  :  :  -525  :  10-5  =  the 
riiie  required. 

31.  All  the  circumstances  comprehended  under  this  kind 
<  'fotatory  motion,  may  be  expressed  by  the  following  rules  : 

*<et  W  express  the  weight  of  a  wheel, 

F,  the  force  acting  upon  the  wheel, 

D,  the  distance  of  the  force  from  the  axis  of  motion, 

G,  the  distance  of  the  centre  of  gyration  from  the 
axis  of  motion, 

t.  the  time  the  force  acts, 

v,  the  velocity  acquired  by  the  revolving  body  in  that 

time. 

Gj<^W^X  v  _  G  X  W  X  v  _  D 

D~X~~rX  32  "  ~  F  X  t   X  32"  ~ 

F  X  D  X  t  X  32  _  F  X  D  X  t  X  32  _ 

W  X  v  G  X  v 

G  X  W  X  v  F  X  D  X  t  X  32  _ 

F  X  D  X  32  ^  *  G  X  W 


GYRATION    AND    ROTATION.  145 

It  is  to  be  observed,  before  applying  these  rules,  that  the 
number  of  turns  of  a  revolving  body  in  a  minute  are  often 
given,  and  it  is  required  to  find  the  velocity  of  feet  per 
second.  A  wheel  of  8  feet  diameter,  for  instance,  makes  12 
revolutions  in  a  minute  ;  how  many  iVet  does  a  nail  in  its 
circumference  pass  over  in  a  second?  Here,  8  x  3-1416 
=  25-1328  feet  the  nail  passes  through  in  one  revolution, 
but  25-1328  X  12  =  301-5936  =  the  feet  it  passes  through 
in  a  minute;  hence,  60)301-5936(5-0265,  the  velocity  in 
ft.  per  second.  The  whole  may  be  expressed  shortly  thus  : 

8  X  3-1416  X  12        „ 

—  =  o-026a. 
60 

Ex.  —  What  must  be  the  weight  of  a  fly-wheel  that  makes 
12  revolutions  in  a  minute,  whose  diameter  is  8  feet,  urged 
by  a  force  of  84  Ibs.  at  its  rim,  acting  for  6  seconds,  the 
distance  of  the  centre  of  gyration  being  3  feet  6  inches  ? 
84  X  4  x  6  x  32 


*5  X  5-0265 

In  a  wheel  which  is  2  tons  weight,  and  12  feet  diameter, 
the  centre  of  gyration  is  6  feet  from  the  centre  of  rotation, 
the  velocity  with  which  this  wheel  moves  is  10  feet  per 
second  ;  what  force  must  be  applied  for  8  seconds,  at  the 
distance  of  3  feet  from  the  centre,  to  generate  that  velocity  ? 


What  is  the  distance  of  the  centre  of  gyration  from  the 
centre  of  motion  of  a  fly-wheel,  the  force  which  moves  the 
wheel  being  2  cwt.,  acting  at  the  distance  of  7  feet  from 
the  centre  of  motion,  and  for  10  seconds,  the  weight  of  the 
wheel  being  2£  tons,  and  its  velocity  8  feet  per  second  ? 
Here  2§  tons  =  50  cwt.  . 

2X7X10X  12        Ml  .    t    ,.  4  , 

-  —  =  Hi  feet,  distance  of  centre  of  gyration. 

oU  X   O 

What  is  the  velocity  acquired  by  a  fly-wheel  acted  upon 
by  a  force  of  84  Ibs.,  at  the  distance  of  4  feet  from  the  axis, 
the  time  in  which  the  force  has  been  acting  is  7  seconds,  the 
weight  of  the  wheel  1A  tons,  and  the  distance  of  the  centre 
of  gyration  5  feet  from  the  centre  of  motion?  Here  lj 
ton  =  30  cwt.  =  3360  Ibs.  ;  therefore, 

84  X  4  X  7  X  32 

—  K  —  oo«n         =  **'"*  ^eet  Per  second»  u16  velocity 
5  X  oooU 

acquired  by  the  wheel. 

13 


146  CENTRAL    FORCES. 


CENTRAL  FORCES. 

1.  INTIMATELY  connected  with  the  foregoing  subject  is 
that  of  central  forces,  the  nature  of  which  maybe  illus- 
trated by  a  very  simple  instance.  When  a  boy  causes  a  stone 
in  a  sling  to  revolve  round  his  hand,  the  stone  is  kept  from 
flying  off  by  the  strength  of  the  string,  which  continually 
draws  the  stone,  as  it  were,  to  the  hand  or  centre  of  motion ; 
but  if  the  string  is  let  go,  or  breaks,  then  the  stone  will  fly 
off  in  a  straight  line,  by  means  of  its  centrifugal  force ; 
the  strength  of  the  string,  while  it  restrains  this  tendency, 
is  called  the  centripetal  force  :  when  both  forces  are  spoken 
of  they  are  jointly  called  central  forces. 

2.  When  a  body  revolves  round  a  fixed  centre,  the  cen- 
tripetal force  may  sometimes  be  the  cohesion  of  the  par- 
ticles of  which  the  body  is  composed,  or  sometimes  it  may 
be  the  power  of  some  attracting  body — such  as  gravity  in 
the  case  of  the  planets. 

3.  In  talking  of  the  angular  velocity  of  a  revolving  body, 
we  mean  not  the  space  which  is  passed  over  in  a  given  time, 
out  the  number  of  degrees,  minutes,  &c.,  that  the  body  de- 
scribes in  a  certain  time,  whether  the  circle  be  large  or 
small.  Thus,  a  body  moving  in  a  circle  of  10  feet  diameter, 
may  have  an  angular  velocity  of  15°  in  a  second,  so  may 
also  another  body  moving  in  a  circle  of  three  feet  diameter  ; 
they  will  complete  their  respective  circles  in  the  same  time, 
but  the  actual  spaces  they  pass  through  are  very  different ; 
that  is,  their  angular  velocities  are  the  same,  but  their  actual 
velocities  are  not. 

4.  The   central  forces   are   as   the  radii   of  the   circles 
directly,  and  the  squares  of  the  times  inversely,  afso  the 
squares  of  the  times   are   as  the  cubes  of  the  distances. 
When  a  body  revolves  in  a  circle  by  means  of  central  forces, 
its  actual  velocity  is  the  same  as  it  would  acquire  by  falling 
through  half  the  radius  by  the  constant  action  of  the  centri- 
petal force.     From  these  facts  the  following  rules  for  cen- 
tral forces  are  derived. 

_  veloc.  of  rev.  body9  X  weight  of  body 

5.  - — .. -.-—. r— , — ~ r-^ — -  =  centrif.  force. 

radius  of  circle  of  revolution  X  32 

„  velocity  of  revol.  body8  x  weight  of  body 

6. J- — rj |L 1  =  radius  of 

centrifugal  force  x  32 

the  circle  of  revolution. 


CENTRAL,    FORCES.  147 

„  centril.  force  x  32  X  rad.  circle 
7.  --  r-  i~  —  "  =  weight  of  the  re- 

veloc.  ot  revolving  body' 

volving  body. 

l/rad.  circle  x  32  x  centrifugal  force\ 
8..   I  -/*•  velocity. 

\  v  weight 

9.  There  will  be  no  difficulty  in  applying  what  has  been 
said  to  practice. 

There  are  two  fly-wheels  of  the  same  weight,  one  of 
which  is  10  feet  diameter,  and  makes  6  revolutions  in  a 
minute  ;  what  must  the  diameter  of  the  other  be  to  revolve 
3  times  in  a  minute  ?  Here  6a  :  3a  :  :  10  :  2-5  =  the 
diameter  of  the  second. 

What  is  the  centrifugal  force  of  the  rim  of  a  fly-wheel, 
its  diameter  being  12  feet,  and  the  weight  of  the  rim  1  ton, 
making  65  turns  in  a  minute  ? 

8  X  3-1416  X  65  =  4Q.84  = 

the  velocity  in  feet  per  second  ;  hence, 
40-848  X  1 


32X6 

the  tendency  to  burst. 

Let  us  employ  the  centre  of  gyration.  —  If  the  fly  above 
mentioned  is  in  two  halves,  which  are  joined  together  by 
bolts  capable  of  supporting  4  tons  in  all  their  positions,  the 
whole  weight  of  the  wheel  is  1  5  tons,  the  circle  of  gyration 
is  5-5  feet  from  the  axis  of  motion  ;  what  must  be  its  velocity 
so  that  its  two  halves  may  fly  asunder?  The  force  tending 
to  separate  the  two  halves  will  be  5  of  the  whole  force  ; 
wherefore,  by  the  rule, 

X  4  *55'8  X  2  =  3°-636  =  the  velocity, 

11  X  3-1416  =  34-5576  =  circumference  of  circle  of  gy- 
ration, wherefore,  34-5576  :  30-636  :  :  60  :  53-191  revolt! 
tions  in  a  minute. 

•  10.  The  steam  engine  governor,  or  conical  pendulum; 
action  the  principle  of  central  forces.  It  is  so  constructed, 
that  when  the  balls  diverge,  or  fly  outwards,  the  ring  on 
the  upright  shaft  is  raised,  and  that  in  proportion  to  the  in- 
crease of  the  velocity,  squared  ;  or,  the  square  roots  of  the 


148  CENTRAL    FORCES. 

d. stances  of  the  ring  from  the  top,  corresponding  to  two 
velocities,  will  be  as  these  velocities. 

Ex. — If  a  governor  makes  6  revolutions  in  a  second, 
when  the  ring  is  16  inches  from  the  top  ;  what  will  be  the 
distance  of  the  ring  when  the  speed  is  increased  to  10  revo- 
lutions in  the  same  time  ?  The  balls  will  diverge  more, 
consequently  the  ring  will  rise  and  the  distance  from  tlie 
top  become  less  ;  therefore,  we  have 

10  :  6  ::  v'  16  or  4  :  2-4; 

which,  squared,  gives  5*76  inches,  the  second  distance  of 
the  ring  from  the  top.     See  Steam  Engine. 

11.  We  shall  elsewhere  introduce  other  particulars  on 
rotation  and  central  forces. 


STRENGTH   OF   MATERIALS,  MACHINES, 
MODELS,   &c. 

MATERIALS  are  exposed  to  four  different  kinds  of  strain : 

1st.  They  may  be  torn  asunder,  as  in  the  case  of  ropes 
ana  stretchers.  The  strength  of  a  body  to  resist  this  kind 
of  strain  is  called  its  Resistance  to  Tension,  or  Absolute 
strength. 

2d.  They  may  be  crushed  or  compressed  in  the  direction 
of  their  length,  as  in  the  case  of  columns,  truss  beams,  &c. 

3d.  They  may  be  broken  across,  as  in  the  case  of  joists, 
rafters,  &c.  The  strength  of  a  body  to  resist  this  kind  of 
strain  is  called  its  Lateral  strength. 

4th.  They  may  be  twisted  or  wrenched,  as  in  the  case 
of  axles,  screws,  &c. 

Extensive  and  accurate  experiments  are  necessary  to 
determine  the  several  measures  of  these  strengths  in  the 
different  materials;  ami  when  this  is  done,  the  subsequent 
calculations  become  comparatively  easy.  We  shall,  there- 
fore, in  the  first  place,  lay  down  the  results  of  the  experi- 
ments of  practical  men. 


STRENGTH    OF    MATERIALS. 


149 


A. 

TABLE    OF    THF    FLEXIBILITY    AND    STRENGTH    OF    TIMBER. 


Name  of  the  Wood. 

u 

E 

s 

c 

Teak, 

818 

9657802 

2462 

15555 

Poon, 

596 

6759200 

2221 

14787 

English  oak, 

598 

3494730 

1181 

9836 

Do. 

435 

5806200 

16.72 

10853 

Canada  oak, 

588 

8595864 

1766 

11428 

Dantzic  oak, 

724 

4765750 

1457 

7386 

Adriatic  oak, 

610 

3885700 

1583 

8808 

Ash, 

395 

6580750 

2026 

17337 

Beech, 

615 

5417266 

1556 

9912 

Elm, 

509 

2799347 

1013 

5767 

Pitch  pine, 

588 

4900466 

1632 

10415 

Red  pine, 

605 

7359700 

1341 

10000 

New  English  fir, 

757 

5967400 

1102 

9947 

Riga  fir, 

588 

5314570 

1108 

10707 

Do. 

3962800 

1051 

Mar  forest  fir, 

588 

2581400 

1144 

9539 

Do. 

403 

3478328 

12.62 

10691 

Larch, 

411 

2465433 

653 

Do. 

518 

3591133 

832 

Do. 

518 

4210830 

1127 

7655 

Do. 

518 

4210830 

1149 

7352 

Norway  spar, 

648 

5832000 

1474 

12180 

NOTE. — The  extensive  use  of  the  above  table  will  be 
shown  hereafter. 

U.  The  ultimate  strength. — E.  Lateral  strength. — S 
Transverse  strength. — C.  Cohesion. 


B. 

T^ble  showing  the  weight  that  will  pull  asunder  a  prism 
one  inch  square. 

Ibs.  Ibs. 

Cast  gold, 22000  '  Bismuth, 29000 

Cast  silver, 41000  I  Good  brass, 51000 

13* 


150 


STRENGTH    OP    MATERIALS. 


Ibs. 

Anglesea  copper,  •  •  •  •  34000 
Swedish  copper,  ••••   37000 

f^nst   ifrvn     ..                                .       ^OOOO 

It*. 

Bar  iron,  ordinary,"   68000 
Do.     Swedish,  ••   84000 

Tt-ir  «tr>f>l      anft     .         ..190000 

COMPOSITIONS 

Gold  5,  copper  1,  • 
Silver  5,  copper  !,• 
Swed.  copper  6,  tin 
Block  tin  3,  lead  1, 
Tin  4,  lead  1,  zinc 

OF 

•-50000 
...48500 
1,.  64000' 
...10200 
1,.  13000 
...45000 

Do.  razor  temper,150000 
Cast  tin,  Eng.  block,     5200 

Dn         n-rsjin    ..                    fi'lOO 

7.\nf.  :...                                          2flOO 

c. 


The  same  from  Rennie  : 


Weight  that  would  tear 
it  asunder  in  Ibs. 


Length  in  feet  that  would 
break  with  its  own  weight 


Cast  steel, 134256 

Swedish  iron,-"."  72064 

English  iron, 55872 

Cast  iron, 19096 

Cast  copper, 19072 

Yellow  brass, 17958 

Cast  tin, 4736 

Castlead,"-- 1824 


39455 

19740 

16938 

6110 

5092 

5180 

1496 

306 


Good  hemp  rope,  ••• 
Do.  one  inch  diam. 


6400  18790 

5026  18790 

D. 


The  cohesive  force  of  a  square  inch  of  iron ;  from  dif- 
ferent experimentists. 


Ibs. 

. 

Ibs. 
...fil  fiOfl 

Dn 

Dn 

...fifVTTP 

..  .  R4QfiO 

Tin 

...  f^^ffR 

,  .  ,(\i  nni 

T)n             

..  .  ^04.79 

Fin 

Dn. 

.    5SOOO 

..1fi9S5 

STRENGTH    OP    MATERIALS.  151 

E. 

Table  of  the  lateral  strength  of  the  following  materials 
one  foot  long,  and  one  inch  square. 


Weight  that  will 
break  them. 

Weight  which  they  CM 
tear  with  safety. 

O-jt    . 

.     20Q  - 

1  1ft  — 

A  rri*>rir»;in   whitfimnp.. 

.      9OR   _ 

C.'.t    - 

F. 

The  force  necessary  to  crush  one  cubic  inch. 

Aberdeen  granite,  blue,  

Very  hard  freestone, 

Black  Limerick  limestone, 

Compact  limestone, 

Craigleith  stone, 

Dundee  sandstone, 

Yorkshire  paving  stone, 

Redbrick, 1817 

Pale  red  brick, 1265 

Chalk, 1127 

Cubes  of  one-fourth  of  an  inch. 

Iron  cast  vertically, 11140 

horizontally, 10110 

Cast  copper, 7318 

Cast  tin, 9.66 

Cast  lead, 483 

Having  made  these  statements,  we  shall  proceed  to  show 
how,  by  the  assistance  of  theoretical  results,  they  may  be 
applied  to  the  wants  of  the  practical  engineer. 

The  absolute  strength  of  ropes  or  bars,  pulled  length 
wise,  is  in  proportion  to  the  squares  of  their  diameters. 
All  cylindrical  or  prismatic  rods  are  equally  strong  in  every 
part,  if  they  are  equally  thick,  but  if  not,  they  will  break 
where  the  thickness  is  least. 

'The  lateral  strength  of  any  beam  or  bar  of  wood,  stone, 
met^il.  &c.,  is  in  proportion  to  its  breadth  X  its  depth8.— • 
In  square  beams  the  lateral  strengths  are  in  proportion  to 
the  cubes  of  the  sides,  and  in  general  of  like-sided  beams 
as  the  cubes  of  the  similar  sides  of  the  section. 


152  STRENGTH    OP    MATERIALS. 

The  lateral  strength  of  any  beam  or  bar,  one  end  being 
fixed  in  the  wall  and  the  other  projecting,  is  inversely  as 
the  distance  of  the  weight  from  the  section  acted  upon ; 
and  the  strain  upon  any  section  is  directly  as  the  distance 
of  the  weight  from  that  section. 

If  a  projecting  beam  be  fixed  in  a  wall  at  one  end,  and 
a  weight  be  hung  at  the  other,  then  the  strain  at  the  end  in 
the  wall,  is  the  same  as  the  strain  upon  a  beam  of  twice 
the  length,  supported  at  both  ends  and  with  twice  the 
weight  acting  on  its  middle.  The  strength  of  a  projecting 
beam  is  only  half  of  what  it  would  be,  if  supported  at  both 
•mds. 

If  a  beam  be  supported  at  both  ends,  and  a  weight  act 
upon  it,  the  strain  is  greatest  when  the  weight  is  in  the 
middle ;  and  the  strain,  when  the  weight  is  not  in  the 
middle,  will  be  to  the  strain  when  it  is  in  the  middle,  as 
the  product  of  the  weight's  distances  from  both  ends,  is  to 
the  square  of  half  the  length  of  the  beam. — Take  any  two 
points  in  a  beam  supported  at  both  ends  ;  call  one  of  these 
points  a  and  the  other  b ;  then  a  weight  hung  at  a  will 
produce  a  strain  at  6,  the  same  as  it  would  do  at  a  if  hung 
at  b. 

In  a  beam  supported  at  the  ends  p c 

A.  and  B  ;  the  strain  at  C,  with  the 

whole  weight  placed  there,  is  to  the  strain  at  C  with  the 
whole  weight  placed  equally  between  C  and  P,  as  AC  is 
to  AP  x  5  PC  ;  and  the  strain  at  C  by  a  weight  placed 
equally  along  AP,  is  to  the  strain  at  C  by  the  same  weight 
placed  on  C,  as  £AP  is  to  AC. 

If  beams  bear  weights  in  proportion  to  their  lengths, 
either  equally  distributed  over  the  beams  or  placed  in  similar 
points,  the  strains  upon  the  beams  will  be  as  their  lengths2. 

If  a  beam  rest  upon  two  supports,  and  at  the  same  time 
be  firmly  fixed  in  a  wall  at  each  end,  it  will  bear  twice  as 
much  weight  as  if  it  had  lain  loosely  upon  the  supports ; 
and  the  strain  will  be  everywhere  equal  between  the 
supports. 

In  any  beam  standing  obliquely,  or  in  a  sloping  direction, 
As  strength  or  strain  will  be  equal  to  that  of  a  beam  of  the 
same  breadth,  thickness,  and  material,  but  only  of  the 
length  of  the  horizontal  distance  between  the  points  of 
support. 

Similar  plates  of  the  same  thickness,  either  supported  at 


STHENGTII    OF    MATERIALS.  153 

the  ends  or  all  round,  will  carry  the  same  weight  either 
uniformly  distributed  or  laid  on  similar  points,  whatever  be 
their  extent. 

The  strength  of  a  hollow  cylinder,  is  to  that  of  a  solid 
cylinder  of  the  same  length  and  the  same  quantity  of  mat- 
ter, as  the  greater  diameter  of  the  hollow  cylinder  is  to  the 
diameter  of  the  solid  cylinder ;  and  the  strength  of  hollow 
cylinders  of  the  same  length,  weight,  and  material,  are  as 
their  greater  diameters. 

The  lateral  strength  of  beams,  posts,  or  pillars,  are  dimi- 
nished the  more  they  are  compressed  lengthwise. 

The  strength  of  a  column  to  resist  being  crushed  is 
directly  as  the  square  of  the  diameter,  provided  it  is  not  so 
long  as  to  have  a  chance  of  bending.  This  is  true  in  metals 
or  stone,  but  in  timber  the  proportion  is  rather  greater  than 
the  square. 

The  strength  of  homogeneous  cylinders  to  resist  being 
twisted  round  their  axes,  is  as  the  cubes  of  their  diameters ; 
and  this  holds  true  of  hollow  cylinders,  if  their  quantities 
of  matter  be  the  same. 

PROBLEMS. 

To  find  the  strength  of  direct  cohesion  : 

Area  of  transverse  section  in  inches  x  measure  of  cohe- 
sion =  strength  in  Ibs.  to  resist  being  pulled  asunder. 

Ex. — In  a  square  bar  of  beech,  3  inches  in  the  side,  we 
have  3  X  3  X  9912  =  89208  Ibs. 

NOTE. — The  measure  of  cohesion  for  timber  is  taken 
from  col.  C,  table  A,  and  for  other  materials,  from  tables  B 
or  C. 

In  a  beam  of  English  oak,  having  four  equal  sides,  each 
side  being  four  inches,  we  have 

4  x  4  x  9836  =  157376  Ibs.,  the  strength. 

In  a  rod  of  cast  steel,  2  inches  broad  and  Ik  inch  thick, 
we  have  2  X  U  X  134256  =  402768  Ibs.,  the  strength. 

What  is  the  greatest  weight  which  an  iron  wire  -£v  of  an 
inch  thick  will  bear  ? 

The  area  of  the  cross  section  of  such  wire  will  be  '007854, 
hence  we  have  -007854  x  84000  =  659-736  Ibs. 

/To  find  the  ultimate  transverse  strength  of  any  beam : 
When  the  beam  is  fixed  at  one  end  and  loaded  at  the 
other  then  the  dimensions  being  in  inches, 


154  STRENGTH    OF    MATERIALS. 

breadth  x  depth2  x  transverse  strength 


=  the  ultimate 


length  of  beam 
transverse  strength. 

NOTE.  —  In  column  S,  Table  A,  will  be  found  the  trans- 
verse strength  of  timber,  and  in  table  E,  that  of  iron,  <fec.  , 
and  let  it  be  observed,  that  when  the  beam  is  loaded  uni- 
formly, the  result  of  the  last  rule  must  be  doubled. 

What  weight  will  break  a  beam  of  Riga  fir,  fixed  at  one 
end  and  loaded  at  the  other,  the  breadth  being  3,  depth  4, 
and  length  60  inches  ? 

886    Ibs. 


What  weight  uniformly  distributed  over  a  beam  of 
English  oak  would  break  it,  the  breadth  being  6,  depth  9, 
and  its  length  12  feet? 

x  a  =  ime  Ibs. 


144 

If  the  number  be  taken  from  table  F,  we  must  use  the 
length  in  feet. 

When  the  beam  is  supported  at  both  ends,  and  loaded  in 
the  centre, 

tabular  value  of  S,  tab.  A  X  depth3  X  breadth  X  4  _ 

length 
the  weight  in  pounds. 

NOTE.  —  When  the  beam  is  fixed  at  one  end  and  loaded 
in  the  middle,  the  result  obtained  by  the  rule  must  be  in- 
creased by  its  half.  When  the  beam  is  loaded  uniformly 
throughout,  the  result  must  be  doubled.  When  the  beam 
is  fixed  at  both  ends  and  loaded  uniformly,  the  result  must 
be  multiplied  by  three. 

Ex.  —  What  weight  will  it  require  to  break  a  beam  of 
English  oak,  supported  at  both  ends  and  loaded  in  the 
middle,  the  breadth  being  6,  and  depth  8  inches,  and  length 
12  feet? 

1672  X  8a  X  6  X  4 


144 

By  using  table  E  : 

depth3  x  breadth  X  tabular  number 
length  in  feet 


PROHLEMS.  155 

Ex. — What  weight  will  a  cast    iron  bar  bear,   10  feet 

ong,  10  inches  deep,  and  2  inches  thick,  laid  on  its  edge  * 

108  x  2  x  1090 


10 

The  same  on  its  broad  side : 
2-  x  10  x  1090 


=  21800  Ibs. 


=  4360  Ibs. 


10 

To  find  the  breadth  to  bear  a  given  weight. 

length  x  weight 

— rr-5-      — rs  =  breadth, 
number  m  table  L  x  depth3 

What  must  be  the  breadth  of  an  oak  beam,  20  feet  long 
and  14  inches  deep,  to  sustain  a  weight  of  10000  Ibs.  ? 

20  X  10000 

— -  =  4-8o  inches  =  the  breadth. 
142  x  209 

To  find  the  length  : 

depth2  X  breadth  x  tabular  number 

-  =  length, 
weight 

In  a  beam  1  ft.  deep  and  4  in.  broad,  the  weight  being 
5000  Ibs. ;  then  we  have,  if  the  beam  be  made  of  Memel  fir, 

123  X  4  X  130 

=  14-97  feet,  length  required. 

5000 

To  find  the  depth  : 

|/         length  X  weight         \  . 

\  \tabular  number  x  breadth/ 

We  wish  to  support  a  weight  of  2000  Ibs.  by  a  beam  of 
American  pine  ;  what  is  its  depth,  its  length  being  20  feet 
and  breadth  4  inches  ? 

2000X20\ 

— -7—  )  =  >/  (145)  =  12  inches,  nearly. 

t>y  x  4 

To  find  the  deflection  of  a  beam  fixed  at  one  end,  and 
loaded  at  the  other  : 

length  of  beam  in  inches3  X  32  x  weight 
f    tab.  numb.  E  (in  table  A)  x  breadth  x  depth3 
flection  in  inches. 

J(OTE. — If  the  beam  be  loaded  uniformly,  use  12  instead 
of  32  in  the  rule. 

If  a  weight  of  300  be  hung  at  the  end  of  an  ash  bar  fixed 


156  STRENGTH    OF    MATERIALS. 

m  a  wall  at  one  end,  and  five  feet  long,  it  being  4  inchei 
square  :  what  is  its  deflection  ? 

603  X  32  X  300 

=  1-23  inches  =  the  deflection. 

6580750  X  4  x  43 

If  the  beam  be  supported  at  both  ends  and  loaded  in  the 
middle : 

length  (in  inches)3  X  weight  . 

tab.  numb*.  (E,  table  A)  X  breadth  x  depth3 

NOTE. — When  the  beam  is  firmly  fixed  at  both  ends,  the 
deflection  will  be  f  of  that  given  by  the  rule. 

Ex. — If  a  beam  of  pitch  pine,  8  inches  broad,  3  inches 
thick,  and  thirty  feet  long,  is  supported  at  both  ends  and 
loaded  in  the  centre  with  a  weight  of  100  Ibs. ;  what  is  its 
deflection  ? 

3603  x  100 

— — =  4-407  inches,  deflection. 

4900466  X  8  x  33 

If  the  beam  had  been  firmly  fixed  at  both  ends,  the  de- 
flection would  have  been 

4-408  X  |  =  2-938  inches. 

If  the  beam  had  been  supported  at  both  ends,  and  loaded 
uniformly  throughout,  the  deflection  would  have  been 

4-408  X  !-  =  2-754. 

To  find  the  ultimate  deflection  of  a  beam  <f  timber 
before  it  breaks  : 

length  (in  inches}8 

-p--^ — 3 — p  =  ultimate  deflection, 
tab.  numb.  U  (table  A)  X  depth 

What  is  the  ultimate  deflection  of  a  beam  of  ash,  1  foot 
broad,  8  inches  deep,  and  40  feet  long  ? 

AQ(\2 

•—- =  72-72lnches,  the  ultimate  deflection. 

To  find  the  weight  under  which  a  column  placed  verti- 
cally will  begin  to  bend,  when  it  supports  that  weight : 
tab.  numb.  E  (table  A)  x  least  thickness3  x  greatest  x  -2056 

length  (in  inches)2 

=  weight  in  pounds. — It  will  be  found  by  the  application 
of  this  rule,  that  it  will  require  40289-22  Ibs.  to  bend  a 
beam  of  English  oak  20  ft.  long,  6  in.  thick,  and  9  in.  broad. 

BEAMS. 

WE  take  the  liberty  here  of  introducing  a  short  extract 
from  Messrs.  Hann  and  Dodds'  Mechanics,  on  the  subject 


BEAMS. 


157 


of  beams.  "  In  the  construction  of  beams,  it  is  necessary 
that  their  form  should  be  such  that  they  will  be  equally 
strong  throughout.  If  a  beam  be  fixed  at  one  end,  and 
loaded  at  the  other,  and  the  breadth  uniform  throughout  its 
length,  then,  that  the  beam  may  be  equally  strong  through- 
out, its  form  must  be  that  of  a  parabola.  This  form  is 
generally  used  in  the  beams  of  steam  engines." 

Dr.  Young  and  Mr.  Tredgold  have  considered  that  it  will 
answer  better,  in  practice,  to  have  some  straight-lined 
figure  to  include  the  parabolic  form  ;  and  the  form  which 
they  propose  is  to  draw  a  tangent  to  the  point  A  of  the 
parabola  ACB. 

To  draw  a  parabola. — 
Let  CB  represent  the 
length  of  the  beam,  and 
AB  the  semi-ordinate,  or 
half  the  base ;  then,  by 
the  property  of  the  para- 
bola, the  squares  of  all 
ordinates  to  the  same 
diameter  are  to  one  an- 
other as  their  respective  abscisses.  Now,  if  we  take  CB 
=  4  feet,  and  AB  =  1  foot,  we  may  proceed  to  apply  this 
property  to  determine  the  length  of  the  semi-ordinates 
corresponding  to  every  foot  in  the  length  of  the  beam,  aa 


follow : — 


CB 

that  is,  48 


ABa 
12a 


CF 
35 


EF8; 

108  =  EF2; 


the  square  root  of  which  is  10'4  nearly  =  EF. 
CG:  GHa; 
24  :  72  =  GH3; 


And  CB  :  ABa 
48  :    123 

the  square  root  of  wh  ch  is  8-5  nearly  =  GH. 
CB:AB9:    CI  :  IK8; 
48  :    129 :     12  :  36  =  IK3; 
the  square  root  of  which  is  6  inches  =  IK. 
Now,  if  we  take  CL  =  6  inches, 
then  CB  :  BB9::  CL  :  LM9; 

48  :    129::      6  :  18  =  LM3; 

the  square  root  of  which  is  4-24,  which  is  very  near  4| 
inches  =»  LM.    Now,  if  any  flexible  rod  be  bent  so  as  just 
to  touch  the  tops  A,  E,  H,  K,  M,  of  the  ordinates,  and  the 
vertex  O,  then  the  form  of  this  rod  is  a  parabola. 
To  draw  a  tangent  to  any  point  A  of  a  parabola: — 
From  the  vertex  G  of  the  parabola  draw  CD  perpendicu 
14 


158  STRENGTH    OF    MATERIALS. 

ar  to  CB,  and  make  it  equal  to  £  AB ;  then  join  AD,  and 
the  right  line  AD  will  be  a  tangent  to  the  parabola  at  the 
point  A ;  that  is,  it  touches  the  parabola  at  that  point.  '  In 
the  same  manner,  we  may  draw  a  tangent  to  the  parabola 
at  any  other  point,  by  erecting  a  perpendicular  at  the  vertex 
equal  to  half  the  semi-ordinate  at  that  point. 

When  a  beam  is  regularly  diminished  towards  the  points 
that  are  least  strained,  so  that  all  the  sections  are  similar 
figures,  whether  it  be  supported  at  each  end  and  loaded  in 
the  middle,  or  supported  in  the  middle  and  loaded  at  each 
end,  the  outline  should  be  a  cubic  parabola. 

When  a  beam  is  supported  at  both  ends,  and  is  of  the 
same  breadth  throughout,  then,  if  the  load  be  uniformly 
distributed  throughout  the  length  of  the  beam,  the  line 
bounding  the  compressed  side  should  be  a  semi-ellipse. 

The  same  form  should  be  made  use  of  for  the  rails  of  a 
wagon-way,  where  they  have  to  resist  the  pressure  of  a 
load  rolling  over  them. 

MODELS. — The  relation  of  models  to  machines,  as  to 
strength,  deserves  the  particular  attention  of  the  mechanic. 
A  model  may  be  perfectly  proportioned  in  all  its  parts  as  a 
model,  yet  the  machine,  if  constructed  in  the  same  propor- 
tion, will  not  be  sufficiently  strong  in  every  part ;  hence, 
particular  attention  should  be  paid  to  the  kind  of  strain  the 
different  parts  are  exposed  to ;  and  from  the  statements 
which  follow,  the  proper  dimensions  of  the  structure  may 
be  determined. 

If  the  strain  to  draw  asunder  in  the  model  be  1,  and  if 
the  structure  is  8  times  larger  than  the  model,  then  the 
stress  in  the  structure  will  be  83  =  512.  If  the  structure  is 
6  times  as  large  as  the  model,  then  the  stress  on  the  struc- 
ture will  be  63  =  216,  and  so  on ;  therefore,  the  structure 
will  be  much  less  firm  than  the  model ;  and  this  the  more, 
as  the  structure  is  cube  times  greater  than  the  model.  If 
we  wish  to  determine  the  greatest  size  we  can  make  a  ma- 
chine of  which  we  have  a  model,  we  have, 

The  greatest  weight  which  the  beam  of  the  model  can 
bear,  divided  by  the  weight  which  it  actually  sustains  =  a 
quotient  which,  when  multiplied  by  the  size  of  the  beam  in 
the  model,  will  give  the  greatest  possible  size  of  the  same 
beam  in  the  structure. 

Ex. — If  a  beam  in  the  model  be  7  inches  long,  and  bear 
a  weight  of  4  Ibs.,  but  is  capable  of  bearing  a  weight  of  26 


SHAFTS  159 

»bs.  ;  what  is  the  greatest  length  which  we  can  make  tha 
corresponding  beam  in  the  structure  ?     Here 


therefore,  6'5  X  7  =  45'5  inches. 

The  strength  to  resist  crushing,  increases  from  a  model 
to  a  structure  in  proportion  to  their  size,  but,  as  above,  the 
strain  increases  as  the  cubes  ;  wherefore,  in  this  case  also, 
the  model  will  be  stronger  than  the  machine,  and  the 
greatest  size  of  the  structure  will  be  found  by  employing 
the  square  root  of  the  quotient  in  the  last  rule,  instead  of 
the  quotient  itself;  thus, 

If  the  greatest  weight  which  the  column  in  a  model  can 
bear  is  3  cwt.,  and  if  it  actually  bears  28  Ibs.,  then,  if  the 
column  be.  18  inches  high,  we  have 


wherefore,  3'464  x  18  =?  62'352  inches,  the  length  of  the 
column  in  the  structure. 

SHAFTS. 

THE  strength  of  shafts  deserves  particular  attention ; 
wherefore,  instead  of  incorporating  it  with  the  general  sub- 
ject, strength  of  materials,  we  have  allotted  to  it  a  separate 
chapter  under  that  head. 

When  the  weight  is  in  the  middle  of  the  shaft,  the  rule  is 

[/ weight  in  Ibs.  X  length  in  feet\ 

3  H—  — — /  =  diameter  in  inches. 

\ v  500 

This  is  to  be  understood  as  the  journal  of  the  shaft,  the 
body  being  usually  square. 

What  is  the  diameter  of  a  shaft  12  feet  long,  bearing  a 
weight  of  6  cwts.,  the  weight  acting  at  the  middle  ? 


672  x  12\  _ 


2-525  inches. 


500 

If  the  weight  be  equally  diffused,  we  have,  the  weight  in 
Ibs.  x  length  ;  extract  the  cube  root  and  divide  by  10;  the 
quotient  is  the  diameter. 

f     Thus,  take  the  last  example,  then  672  x  12  =  8064; 

'  the  cube  root  of  which  is  20*05,  which  divided  by  10  gives 
2'005,  the  diameter  of  the  shaft. 


16C  STRENGTH    OP    MATERIALS. 

If  a  cylindrical  shaft  have  no  other  weight  to  sustain  be- 
sides its  own,  the  rule  is,  v/(-007  X  length3)  =  diameter: 
thus,  if  a  shaft  having  only  the  stress  of  its  own  weight  be 
10  feet  long; 
v/('007xl03)  =  2-645  the  diameter  of  the  shaft  in  inches. 

For  a  hollow  shaft  supporting  so  many  times  its  own 
weight,  we  have 

I/-012  x  length3  x  No.  times  its  own  weight\ 
\  v  1  +  inner  diameter2  / 

outer  diameter  in  inches. 

For  wrought  iron  shafts  find  the  diameter  by  the  forego 
ing  rules,  which  apply  to  cast  iron,  then  multiply  by  -935, 
and  for  oak  shafts  the  multiplier  is  1-83,  and  for  fir  1-716. 

Ex. — What  is  the  diameter  of  a  cast  iron  shaft  12  feet 
long,  and  the  stress  it  bears  being  twice  its  own  weight? 
Here  we  have, 

v'  (-012  X  123  x  2)  =  6-44  inches. 
For  wrought  iron,  using  the  multiplier, 
6-44  x  '935  =  6-0215, 
and  for  oak,  using  the  multiplier, 

6-44  x  1-83  =  11-3852, 
and  for  fir,  we  have 

6-44  X  1-716  =  11-05104. 

A  rule  often  used  in  practice,  though  by  no  means  a  cor- 
rect one,  for  determining  the  diameter  of  shafts  is  this. 
The  cube  root  of  the  weight  which  the  shaft  bears  taken  in 
cwts.  is  nearly  the  diameter  of  the  shaft  in  inches.  It  will 
be  found  safe  in  practice,  to  add  one-third  more  to  this 
result. 

If  a  cast  metal  shaft  has  to  bear  a  weight  of  1|  ton,  that 
is,  30  cwts.,  then  we  have, 

#  30  =  3-107  inches  by  this  rule  ; 

and  supposing  it  12  feet  long,  we  will  apply  the  other  rule, 
we  have, 

3360  X  12\ 

—wo—)  = 4'319- 

We  have  now  considered  the  strength  of  shafts,  so  far  as 
regards  their  power  to  resist  lateral  pressure  by  weight  act- 
ing on  them ;  we  have  now  to  consider  their  power  to 
resist  torsion  or  twisting. 


V/ J  llll 

ifc 


SHAFTS.  161 

For  cylindrical  shafts,  we  have, 

240  X  No.  of  horses'  power  \ 
''No.  of  revolutions  in  a  minute/ 
ihe  diameter  of  the  shaft  in  inches. 

This  rule  is  for  cast  iron  ;  and  it  may  be  used  for 
wrought  iron*by  multiplying  the  result  by  -963,  or  for  oak 
oy  2-238,  or  for  fir  by  2-06. 

If  the  shaft  belong  to  a  7  horse  power  engine,  and  the 
etrap  turns  11^  times  in  a  minute, 

3   (  — —-= — }  =  5-267  inches  diameter  for  cast  iron. 
\  \     11-5     / 

For  fir,  5-267  X  2*06  =  10-85. 

For  oak,  5-267  X  2*38  =  12-535. 

And  for  wrought  iron,  5-267  X  -963  =  5-0719. 

NOTE. — This  rule  comes  from  the  best  authority,  anc 
gives  perfectly  safe  results,  though  some  employ  340,  in 
stead  of  240,  as  a  multiplier,  which  gives  a  greater  diameter 
to  the  shaft.  We  may  compare  the  two : 

==  5-916, 


whereas  the  other  was  5-267 — something  more  than  half  an 
inch  of  difference. 

It  is  to  be  remembered,  that  these  rules  relate  to  the 
iihafts  of  first  movers,  or  the  shafts  immediately  connected 
with  the  moving  power.  But  these  shafts  may  communi- 
cate motion  to  other  shafts,  called  second  movers,  and  these 
again  to  others,  called  third  movers,  and  so  on.  The  dia- 
meters of  the  second  movers  may  be  found  from  those  of 
Ihe  first,  by  multiplying  by  -8,  and  those  of  the  third 
movers,  by  multiplying  by  -793,  thus,  if  the  diameter  of 
the  first  mover  be  5-267,  then  that  of  the  second  will  be 
5-267  X  -8  =  4-2136,  and  that  of  the  third  mover  will  be 
5-267  X  -793  =  4-1767. 

One  material  may  resist,  much  better  than  another,  one 

kind  of  strain  ;  but  expose  both  to  a  different  kind  of  strain, 

and  that  which  was  weakest  before  may  now  be  the  strong- 

"est.    This  may  be  illustrated  in  the  case  of  cast  and  wrought 

icon.   The  cast  iron  is  stronger  than  the  wrought  iron  when 

exposed  to  twisting  or  torsional  strain,  but  the  malleable 

iron  is  the  stronger  of  the  two  when  they  are  exposed  to 

14* 


162 


STRENGTH    OF    MATERIALS. 


lateral  pressure.  We  shall  subjoin  a  few  results  of  experi- 
ments on  the  weight  which  was  necessary  to  twist  bars  j 
close  to  the  bearings. 


Oast  metal,  .............  9 

Do.  vertical  cast,  .....  10 

Cast  steel,  ..............  17 

Shear  steel......  .......  17 

Blister  steel,  ...........  16 


oz. 

17 

10 

9 

1 

11 


English  iron  wrought,.  10 
Swedish  iron  wjought,  9 

Hard  gun  rnetal, 5 

Brass  bent, 4 

Copper  cast, 4     5 


lb.     01 

2 

8 
0 
11 


It  would  appear  that  the  strength  of  bodies  to  resist  torsion 
is  nearly  as  the  cubes  of  their  diameters. 

REMARKS. — The  rules  and  statements  we  have  now  given 
will  often  find  their  application  in  the  practice  of  the  engi- 
neer. On  the  proper  proportioning  of  the  magnitude  of 
materials  to  the  stress  they  have  to  bear,  depends  much  of 
the  beauty  of  any  mechanical  structure ;  and,  what  is  of  fai 
greater  moment,  its  absolute  security.  We  will,  in  the  Ap« 
pendix  to  this  book,  give  some  examples  of  the  application 
of  these  principles  to  practice. 

TABLE  OF  THE  DIAMETERS  OF  SHAFT  JOURNALS. 


10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

5 

5-9 

4-7 

4-1 

3-7 

3-5 

3-3 

3-1 

3-0 

2-9 

2-7 

6 

6-3 

5-0 

4-4 

4 

3-7 

3-5 

3-4 

3-2 

3 

2-9 

7 

6-6 

5-2 

4-6 

4-2 

3-9 

3-6 

3-5 

3-4 

3-3 

3-1 

8 

6-9 

5-5 

4-8 

4-4 

4-1 

3-9 

3-7 

3-5 

3-4  3-3 

9 

7-2 

5-7 

5 

4-5 

4-2 

4 

3-7 

3-6 

3-5 

3-4 

10 

7-4;  5-9 

5-2 

4;7 

4-4 

4-1 

3-9 

3-7 

3-6 

3-5 

15 

8-5 

7-0 

6-0 

5-5 

5-1 

4-6 

4-5 

4-3 

4-2 

4-0 

20 

9-3 

7-4 

6-6 

5-9 

5-6 

5-2 

5-0 

4-6 

4-5 

4.4 

30 

10-7 

8-4 

7-4 

6-9 

6-5 

5-9 

5-7 

5-5 

5-2 

5-0 

40 

11-7  9-5 

8-3 

7-4 

6-9 

6-6 

6-2 

5-9 

5-7 

5-6 

50 

12-610-0 

9-0 

8-0 

7-4 

7-2 

6-8 

6-5 

6-2 

5-9 

60 

13-610-8 

9-3 

8-6 

7-7 

7-4 

7-2 

6-8 

6-7 

6-4 

In  the  preceding  table  of  the  diameters  of  the  shafts  of  first 
movers,  the  number  of  horses'  power  of  the  engine  is  given 
in  the  left-hand  column,  and  the  number  of  revolutions  the 
shaft  makes  in  a  minute  is  given  in  the  top  column.  Then, 
to  use  the  table,  we  have  only  to  look  for  the  power  of  the 
engine  in  the  side  column,  and  the  number  of  turns  the  shaft 


JOISTS    AND    ROOFS.  163 

makes  in  a  minute  in  the  line  which  runs  across  the  top, 
and  where  these  columns  meet  will  be  found  the  diameter 
of  the  shaft  in  inches.  The  table  is  constructed  for  cast 
iron,  and  first  movers ;  the  rules  for  finding  the  second  and 
third  have  been  given  above,  as  also  for  finding  equally 
strong  shafts  of  other  materials. 

This  table  answers  for  first  movers  only.  It  may,  how- 
ever, be  made  to  give  results  for  second  and  third  movers, 
by  using  the  multipliers  for  that  purpose,  formerly  given. 

What  is  the  diameter  of  the  journal  of  the  shaft  of  the 
first  mover  in  a  30  horse  power  engine,  the  shaft  making 
40  revolutions  in  a  minute?  Here,  by  looking  in  the  table, 
in  the  side  column  of  horses'  power,  w<i  find  30,  and  in  the 
top  column  of  revolutions,  we  find  40,  and  whert*  these 
columns  meet,  we  find  6-9  =  the  diameter  of  the  first 
mover,  in  inches ;  wherefore,  the  second  mover  of  this 
power  and  velocity  will  be  =  6-9  X  '8  =  5-52  inches  ;  and, 
in  like  manner,  the  third  mover  will  be  =  6-9  x  "64  = 
4'416  inches  =  the  diameter  of  the  third  mover  to  the  same 
power  and  speed. 

JOISTS    AND    ROOFS. 

JOISTS  should  increase  in  strength  in  proportion  to  the 
squares  of  their  lengths ;  for  instance,  a  joist  10  feet  long 
should  be  four  times  as  strong  as  another  joist  8  feet  long, 
similarly  situated;  because  8* :  16*:  :  1  :  4.  From  what 
has  been  previously  stated,  it  will  easily  appear,  that  the 
stress  on  a  beam  or  joist  supported  at  both  ends,  increases 
towards  the  middle,  where  it  is  greatest ;  it  therefore  fol- 
lows, that  a  beam  should  be  strengthened  in  proportion  to 
the  increasing  strain  ;  and,  as  it  would  not  be  easy  to  add 
to  the  thickness  of  a  beam  towards  the  middle,  which  would 
destroy  the  levelness  of  the  floor,  a  good  substitute  may  be 
to  fasten  pieces  to  the  sides  of  the  joist,  and  thus  increase 
its  breadth ;  thus  causing  the  beam  to  taper,  in  breadth, 
from  the  centre  to  the  ends.  In  this  way  joists  may  be 
made  much  stronger  than  they  usually  are  of  the  same 
'length,  and  out  of  the  same  quantity  of  timl  er.  It  may 
^Iso  be  observed,  that  joists  are  twice  as  strong  when  firmly 
fixed  in  the  wall,  as  when  loose ;  but  it  is  to  be  remarked, 
th%t  they  have,  when  fixed,  a  far  greater  tendency  to  shake 
the  wall.  It  is  also  to  be  remarked,  that  a  joist  is  four  times 
stronger  when  supported  in  the  middle. 


164  JOISTS    AND    ROOFS. 

If  the  letter  L  represent  the  length  of  some  known  joist, 
whose  strength  has  been  tried,  and  D  its  depth,  and  T  its 
thickness  ;  and  if  another  joist  is  required  of  equal  strength 
with  the  former,  when  similarly  situated  ;  whose  length  is 
represented  by  /,  its  depth  by  c/,  and  its  thickness  by  t;  we 
have  the  following  rules  : 

x  l  ,     I/D  •'  x  T  x  1 


Da  x  /'  X  T  ..       /</*  x  /  x  I 


D'xT 

If  a  joist  30  feet  long,  1  foot  deep,  and  3  inches  thick,  be 
sufficient  in  one  case,  what  must  the  depth  of  a  beam  be, 
similarly  placed,  whose  length  is  15  feet,  its  depth  and 
thickness  bearing  the  same  proportion  to  each  other,  as  in 
the  former  beam  ?  Here,  by  the  first  theorem,  we  have, 

=  '6298  fect  =  7'55  inohes 


the  depth  ;  and  therefore  12  depth  :  3  thickness  :  :  7*55  : 
1-88  the  breadth. 

If  the  given  beam  be,  as  in  the  last  example,  12  inches 
deep,  3  thick,  and  30  feet  long,  and  the  required  beam,  of 
the  same  strength,  is  8  inches  deep,  and  6  inches  thick,  then 
by  the  4th  we  have, 


If  a  joist,  whose  length  is  30  feet,  depth  12  inches,  and 
thickness  8,  is  given,  to  find  the  depth  of  another  of  equal 
strength,  only  6  inches  thick,  and  28-28  feet  long?—  Here, 
by  the  2d,  we  have, 

f!2a  x  3  X  2S-283 
\l  --  6  x  SO  --  =     incnes>  l"e  depth. 

To  find  the  thickness  from  the  same  circumstances,  we 
nave  by  the  3J, 

128  x  2S-283  x  3 
—  go  v.  or>a  -  ==  ^  inches,  the  thickness. 

O     X  oU 

The  same  remarks  hold  true  to  a  certain  extent  in  roof- 
ing. A  high  roof  is  both  heavier  and  more  expensive  than 
d  low  roof,  as  they  will  always  be  as  the  squares  of  the 
lengths  of  the  couple-legs,  so  far  as  the  scantling  is  con- 
cerned ;  but  the  slates  and  other  materials  increase  in  weight 


WHEELS.  165 

ind  expense  as  the  length  of  the  couple -legs  simply.  High 
roofs  have,  however,  the  advantage  of  being  less  severe 
upon  the  walls  than  low  ones,  that  is  to  say,  so  far  as  a 
tendency  to  push  out  the  walls  is  concerned.  To  obtain 
the  length  of  the  rafter  from  that  of  the  span,  a  common 
rule  is  to  multiply  the  span  by  -66,  which  gives  the  length 
of  the  rafter;  thus,  14  feet  of  span  gives  14  x  '66  =  9-24 
feet,  the  length  of  the  rafter. 

NOTE. — The  numbers  in  the  tables  of  the  strength  of 
materials  are  such  as  will  break  the  bodies  in  a  very  short 
time  ;  the  prudent  artist,  therefore,  will  do  well  to  trust  no 
more  than  about  one-third  of  these  weights ;  also  great 
allowance  must  be  made  for  knotty  timber,  and  such  as  is 
sawn  in  any  part  across  or  obliquely  to  the  fibres. 


WHEELS. 

IN  page  136  we  promised  again  to  resume  the  subject  of 
•wheel-work  ;  and  we  now  proceed  to  consider,  in  the  first 
place,  the  formation  of  the  teeth  of  wheels. 

A  Cog-wheel  is  the  general  name  for  any  wheel  which 
has  a  number  of  teeth  or  cogs  placed  round  its  circumference. 

A  Pinion  is  a  small  wheel  which  has,  in  general,  not 
more  than  12  teeth  ;  though,  when  two  toothed  wheels  act 
upon  one  another,  the  smallest  is  generally  called  the  pinion ; 
so  is  also  the  trundle,  lantern,  or  wallower. 

When  the  teeth  of  a  wheel  are  made  of  the  same  material 
and  formed  of  the  same  piece  as  the  body  of  the  wheel,  they 
are  called  teeth  ;  when  they  are  made  of  wood  or  some  other 
material,  and  fixed  to  the  circumference  of  the  wheel,  they 
are  called  cogs ;  in  a  pinion  they  are  called  leaves ;  in  a 
trunflle,  staves. 

The  wheel  which  acts  is  called  a  leader,  or  driver  ;  and 
the  wheel  which  is  acted  upon  by  the  former  is  called  a 
foBower,  or  the  driven. 

When  a  wheel  and  pinion  are  to  be  so  formed  that  the 
.pinion  shall  revolve  four  times  for  the  wheel's  once,  then 
they  must  be  represented  by  two  circles,  whose  diameters 
a/e  to  one  another,  as  4  to  1.  When  these  two  circles  are 
so  placed  that  they  touch  each  other  at  the  circumferences, 
then  the  line  drawn  joining  their  centres,  is  called  the  lina 


166  WHEELS. 

of  centres,  and  the  radii  of  the  two  circles  the  proportions* 
radii. 

These  circles  are  called,  by  mill-wrights  in  general, 
pitch-lines. 

The  distances  from  the  centres  of  two  circles  to  the  ex- 
tremities of  their  respective  teeth,  are  called  the  real  radii, 
and  the  distances  between  the  centres  of  two  contiguous  teeth 
measured  upon  the  pitch-line,  is  called  the  pitch  of  the  wheel. 

Two  wheels  acting  upon  one  another  in  the  same  plane, 
are  called  spur  geer.  When  they  act  at  an  angle,  they  are 
called  bevel  geer. 

Teeth  of  wheels  and  leaves  of  pinions  require  great  care 
and  judgment  in  their  formation,  so  that  they  neither  clog 
the  machinery  with  unnecessary  friction,  nor  act  so  irre- 
gularly as  to  produce  any  inequalities  in  the  motion,  and  a 
consequent  wearing  away  of  one  part  before  another.  Much 
has  been  written  on  this  subject  by  mathematicians,  who 
seem  to  agree  that  the  epicycloid  is  the  best  of  all  curves 
for  the  teeth  of  wheels.  The  epicycloid  is  a  curve  differing 
from  the  cycloid  formerly  described,  in  this,  that  the  gene- 
rating circle  instead  of  moving  along  a  straight  edge,  moves 
on  the  circumference  of  another  circle. 

The  teeth  of  one  wheel  should  press  in  a  direction  per- 
pendicular to  the  radius  of  the  wheel  which  it  drives.  As 
many  teeth  as  possible  should  be  in  contact  at  the  same 
time,  in  order  to  distribute  the  strain  amongst  them  ;  by 
this  means  the  chance  of  breaking  the  teeth  will  be  dimi- 
nished. During  the  action  of  one  tooth  upon  another,  the 
direction  of  the  pressure  should  remain  the  same,  so  that 
the  effect  may  be  uniform.  The  surfaces  of  the  teeth  in 
working  should  not  rub  one  against  another,  and  should 
suffer  no  jolt  either  at  the  commencement  or  the  termination 
of  their  mutual  contact.  The  form  of  the  epicycloid  satis- 
fies all  these  conditions ;  but  it  is  intricate,  and  the  involute 
of  the  circle  is  here  substituted,  as  satisfying  equally  these 
conditions,  and  as  being  much  more  easily  described. 

Take  the  circumference  ABC  of  the 
wheel  on  which  it  is  proposed  to  raise 
the  teeth,  and  let  a  be  a  point  from 
which  one  surface  of  one  tooth  is  to 
spring,  then  fasten  a  string  at  B,  such 
that  when  stretched  and  lying  on  the 
circumference  shall  reach  to  a/  fix  a 


WHEELS.  lb'7 

pencil  at  a,  and  keeping  the  string  equally  tense,  move 
the  pencil  outwards,  and  it  will  describe  the  involute  of  the 
circle  which  will  form  the  curve  for  one  side  of  the  tooth. 
Fasten  the  string  at  B  so  that  its  end,  to  which  the  pencil 
is  fixed,  be  at  the  point  from  which  the  other  face  of  the 
tooth  is  to  spring — and  proceed  as  above ;  then  the  curve 
of  the  other  side  of  the  tooth  will  be  formed  ;  and  the  figure 
of  one  of  the  teeth  being  determined,  the  rest  may  be  traced 
from  it. 

The  teeth  of  the  pinion  are  formed  in  like  manner. 

The  observation  of  practical  men  has  furnished  us  with 
a  method  of  forming  teeth  of  wheels,  which  is  found  to  an 
swer  fully  as  well  in  practice  as  any  of  the  geometrical 
curves  of  the  mathematician. 

We  have  the  pattern  here  of 
the  segment  of  a  wheel  with 
cogs  fixed  on  in  their  rough 
state,  and  it  is  required  to  bring 
them  to  their  proper  figure : 
they  are  consequently  understood  to  be  much  larger  than 
they  are  intended  to  be  when  dressed.  The  arc  6,  6,  is  the 
circumference  of  the  wheel  on  which  the  bottoms  of  the 
teeth  and  cogs  rest.  Draw  an  arc  a,  a,  on  the  face  of  the 
teeth  for  the  pitch  line  of  their  point  of  action  ;  draw  also 
rf,  rf,  for  their  extremities  or  tops.  When  this  is  done,  the 
pitch  circle  is  correctly  divided  into  as  many  equal  parts  as 
there  are  to  be  teeth.  The  compasses  are  then  to  be  opened 
to  an  extent  of  one  and  a  quarter  of  those  divisions,  and  with 
this  radius  arcs  are  described  on  each  side  of  every  division 
on  the  pitch  line  a,  a,  from  that  line  to  the  line  d,  d.  One 
point  of  the  compasses  being  set  on  c,  the  curve  f,  g,  on 
one  side  of  one  tooth,  and  o,  n,  on  the  other  sides  of  the 
other  are  described.  Then  the  point  of  the  compasses  being 
set  on  the  adjacent  division  k,  the  curve  /,  m,  will  be  de- 
scribed :  this  completes  the  curved  portion  of  the  tooth  e. 
The  remaining  portion  of  the  tooth  within  the  circle  a,  a, 
is  bounded  by  two  straight  lines  drawn  from  g  and  m  to- 
wards the  centre.  The  same  being  done  to  the  teeth  all 
round,  the  mark  is  finished,  and  the  cogs  only  require  to  be 
dressed  down  to  the  lines  thus  drawn. 
/  It  will  be  easy  to  determine  the  diameter  of  any  wheel 
having  the  pitch  and  number  of  teeth  in  that  wheel  given 
Thus,  a  wheel  of  54  teeth  having  a  pitch  of  3  inches,  we 


163 


WHEELS. 


have   54 
quently, 


X  3  =    162  inches,  the  circumference,   conse- 


162 


—  =  51-5  inches  diameter,  nearly. 


3-1416 
or  about  4  feet  3£  inches. 

In  the  following  table  we  have  given  the  radii  of  wheels 
of  various  numbers  of  teeth,  the  pitch  being  one  inch.  To 
find  the  radius  for  any  other  pitch,  we  have  only  to  multiply 
the  radius  in  the  table  by  the  pitch  in  inches,  the  product 
is  the  answer.  Thus  for  30  teeth  at  a  pitch  of  3|  inches, 
we  have  4-783  x  3-5  =  16-74  inches,  the  radius. 


0 

1    2 

3 

4 

5 

6 

7 

8 

9 

10 

1-668 

1-774  1-932 

2-089 

2-247 

2-405 

2-563 

2-721 

2-879 

3-038 

20 

3-196 

3355  3-513 

3-672 

3-830 

3-989 

4-148 

4-307 

4-465 

4-624 

30 

4-783 

4-942  5-101 

5-260 

5-419 

5-578 

5-737 

5-896 

6-055 

6-214 

40 

6-373 

6-532  6-643 

6-850 

7-009 

7-168 

7-327 

7-486 

7-695 

7-804 

50 

7-963 

8-122  8-231 

8-440 

8-599 

8-753 

8-962 

9-076 

9-235 

9-399 

60 

9-553 

9-7121  9-872 

10-031 

10-190 

10-349 

10-508 

10-662 

10-826 

10-935 

70 

11-144 

11-303  11-463 

11622 

11-731 

11-940 

12-099 

12-758 

12-417 

12-676 

80 

12-735 

12-895  13-054 

13-213 

13370 

13531 

13690 

13-849 

14-008 

14-168 

90 

14-327 

14-4H6  14-645 

14-804 

14-963 

15-122 

15281 

15-441 

15-600 

15-759 

100 

15-918 

16-072  16-236 

16-305 

16-554 

16713 

16-873 

17-032 

17-191 

17-350 

110 

17-504 

17-6U8  17-987 

17-827 

18-146 

18-305 

18-464 

18-623 

18-782 

18-941 

120 

19-101 

19-260  19-419 

19-578 

19-737 

19-896 

20-055 

20-2  f4 

20-374 

20-533 

130 

20-692 

20-851  21-010 

21-169 

21-328 

21-48S 

21-647 

21-806 

21-460 

22-124 

140 

22-283 

22-44222-602 

22-761 

22920 

23074 

23-238 

23397 

23-556 

23-716 

150 

23875 

24-03424-193 

24-352 

24-511 

24620 

24-830 

24-989 

25-148 

25-307 

160 

25-466 

25-625  25-784 

25-944 

26-103 

26-262 

26-421 

26-580 

26-739 

26-894 

170 

27-058 

•27-217  27-376 

27-535 

27-694 

27-853 

27-931 

28-172 

28-331 

28-490 

180 

28-699 

28-80828-967 

29-126 

29-286 

29-445 

29-604 

29-763 

29-922 

30-086 

190 

30-241 

30-400  30-559 

30-718 

30-S77 

31-036 

31-196 

31-355 

31-514 

31-673 

200 

31-832 

31-992  32-150 

32-310 

32-469 

32-628 

32-787 

32-846 

33-105 

33-264 

210 

33-424 

3358333-742 

33-901 

34-060 

34-219 

34278 

34-537 

34-697 

34-856 

220 

35-015 

35-174  35-333 

35-492 

35-652 

35-811 

35-970 

36-129 

36-288 

36-447 

230 

36-607 

36-76636-925 

37-084 

37-243 

37-402 

37-561 

37-720 

37-880 

38-039 

240 

38-198 

38-357  38-516 

38-725 

38-835 

38-994 

39-153 

39312 

39-471 

39-631 

250 

39-790 

39-949!40-108 

40-262 

40-426 

40-585 

40-744 

40-904 

41-063 

41-222 

260 

41-381 

41-54U41-699 

4i-858 

42-019 

42-177 

42-336 

42-495 

42-654 

42-813 

270 

42-973 

43-13243-291 

43-450 

43-609 

43-768 

43-927 

44-087 

44231 

44-405 

280 

44-564 

44-723  44-882 

45-042 

45201 

45360 

45519 

45-678 

45837 

45-996 

290 

46-156146  315,46-474 

46-633 

•16-792 

46-751 

47-111 

47-270 

47-429 

47-583 

1     1 

This  will  be  found  a  very  useful  table  in  abridging  calcu- 
lation,— for  instance,  if  we  wish  to  find  the  radius  of  a  wheel 
having  132  teeth,  we  look  for  130  at  the  left-hand  side 
column,  and  2  at  the  top,  and  where  these  columns  meet, 
we  find  the  number  21-010,  which,  if  the  pitch  of  the 
wheel  be  2£  inches,  we  multiply  by  2|. 


WHKKl.s.  169 

21-010  x  2-5  =  52-525  inches,  radius  of  required  wheel. 
An  easy  practical  rule  Cor  the  same  purpose  is  the  fol- 
lowing : — 

Take  the  pitch  by  a  pair  of  compasses,  and  lay  it  off  on 
a  straight  line,  seven  times,  divide  this  line  into  eleven 
equal  parts  ;  each  will  be  equal  to  four  ot'  the  radius,  which 
is  supposed  to  consist  of  as  many  parts  as  the  wheel  has 
teeth. 

Let  the  pitch  be  two  inches,  and  the  number  of  teeth  60 
t-ien  the  diagram  will  show  how  to  lay  it  down. 

1 2         :*         4567 

i~~2T~3~~4"  ~5~6     7.    8     9      10     11 
C— D 

4,  8,   12,   16,  &c. 

The  upper  line  is  the  pitch  laid  off  seven  times,  and 
forming  AB,  which  is  divided  into  11  equal  parts,  one 
of  which,  CD,  being  repeated  for  every  four  teeth  in 
the  wheel,  that  is,  in  this  case,  fifteen  times,  will  give  the 
radius. 

The  same  may  be  done  by  calculation,  going  by  the 
principles  of  the  rule,  thus, 

2x7  =  14,  then—  =   1-272,  which  divided  by  4 

gives 0-318   =   the  value  of    —  of  the    radius ; 

4  60 

wherefore,  -318  x  60  =  19-08. 
By  the  table  we  have, 

9-552  X  2  =  19-104, 
the  difference  in  the  two  results  being 

19-104 —  19-08  =  -024,  or  twenty-four  thousandth 
parts  of  an  inch. 

Reversing  the  operation,  let  it  be  required  to  find  the 
pitch,  the  radius  of  the  wheel  being  19-104,  and  number 
of  teeth  60. 

We   have  -—-—*•   -318,    then   -318  x  4  =  1-272, 
60 

and  1-272  X  11  =  13-992.  Now  this  is  the  whole  line  AB, 

13*992 

and  therefore,  =  1-998,  which  is  so  verv  nearly 

/  7 

two  inches,  the  difference  being  2 —  1-998  =  -002  oi  an 

inch,  we  ought  in  practice  to  take  two  as  the  pitch. 

15 


170  WHEELS. 

A  little  reflection  on  the  part  of  the  reader  will  show 

*?  11  »6Qfi 

that  since  —  =  -636,  and  —  =  1-571,  and——  =  *159 
117  4 

Tve  have, 

(1)  pitch  x  '159  x  number  of  teeth  =  radius. 
/n\  radius 

'  number  of  teeth  x  '159  = 

radius 

(3)     -  —  ;  -  TTT:  =  number  of  teeth. 
v  '  pitch  X  '159 

Thus, 

(1)  2  X  '159  X  60  =  19-08  =  radius, 
19 


19 

(3)  ^  -  ;W  =  60  =  number  of  teeth. 
v  '  2  x  -159 

NOTE.  —  The  number  -16  may  be  employed  instead  of 
159,  being  easily  remembered.  These  rules  are  approxi- 
mate, and  the  error  diminishes  as  the  number  of  teeth  in- 
creases. The  true  pitch  is  a  straight  line,  but  these  rules 
give  it  an  arc  of  the  circle,  which  passes  through  the  centre 
of  the  teeth,  whereas  it  should  be  the  chord  of  the  arc. 

An  eminent  writer  on  clock-work  gives  the  following 
rules  regarding  wheels  and  pinions  :  — 

(A)  As  the  number  of  teeth  in  the  wheel  +  2-25, 
Is  to  the  diameter  of  the  wheel, 

So  is  the  number  of  teeth  in  the  pinion  +  1-5, 
To  the  diameter  of  the  pinion. 

A  wheel  being  12  inches  diameter,  having  120  teeth, 
drives  a  pinion  of-20  leaves  ;  wherefore, 

120  +  2-25  =  122-25  and  20  +  1-5  =  21-5, 
Then  122-25  :  12  :  :  21-5  :  2^104  =  the  diameter  of  the 
pinion. 

(B)  As  the  number  of  teeth  in  the  wheel  +  2-25, 
Is  to  the  wheel's  diameter, 

So  is  3  (teeth  in  wheel  -f  leaves  in  pinion) 
To  the  distance  of  their  centres. 

A  wheel's  diameter  being  3'2  inches,  number  of  teeth  96, 
the  leaves  in  the  pinion  being  8,  then, 

104 
96  +  2-25,  =  98-25  and  £  (96  -f  8)  =  -  -  =  52. 


171 

Hence,  98-25  :  3'2  : :  52  :  1-6936  =  the  distance  which 
the  centres  ought  to  have. 

The  strength  of  wheels  is  a  subject  which  has  occupied 
the  attention  of  the  most  eminent  practical  engineers,  but 
the  rules  they  have  given  us  are  entirely  empirical,  that  is 
to  say,  the  result  of  experiment. 

The  strength  of  the  teeth  will  vary  with  the  velocity  of 
the  wheel,  the  strength  in  horses'  power  at  a  velocity  ol' 
2-27  feet  per  second,  will  be 

breadth  of  the  tooth  x  its  thickness" 

-  =  power, 
length  of  tooth 

Required  the  strength  in  horses'  power  of  a  tooth  4  inches 
broad,  1-3  inches  thick,  and  1-6  inches  long,  at  a  velocity 
of  2-27  feet  per  second, — here  we  have 

4  X  l"3a 

— — —  =  4-225,  the  horses'  power  at  a  velocity  of  2  27. 
1'6 

The  power  at  any  other  velocity  may  be  found  by  pro- 
portion, thus  the  same  at  6  feet  per  second. 

2-27  :  6  ::  4-225  :  11-1  =  horses'  power  a*  i,  velocity 
of  6  feet  per  second. 

The  thickness  of  a  tooth  x  2-1  =  the  pitch 

The  thickness  of  a  tooth  x  1'2  =  length. 

Ex. — The  thickness  of  a  tooth  being  l£  in-  hos,  then  we 
have 

1-5  X  2-1  =  3-15  =  the  pitch. 
1-5  X  1-2  =  1-8  =  the  length 

The  breadth  in  practice  is  usually  2'5  times  the  pitch. 

The  arms  of  wheels  generally  taper  from  the  axle  to  the 
rim,  because  they  sustain  the  greatest  stress  towards  the 
axle.  It  is  obvious,  that  the  more  numerous  the  arms  of  a 
wheel  are,  they  each  suffer  a  proportionately  less  strain,  as 
the  resistance  will  be  diffused  over  a  greater  number. 
The  power  acting  at  the  rim  X  length  of  :rm3 

number  of  arms  X  2656  X  0-1 
and  cube  of  depth. 

Ex. — If  the  force  acting  at  the  extremity  of  the  arm  of  a 
wheel  be  16  cwt.;  the  radius  of  the  wheel  being  5  feet,  and 
the  number  of  arms  6,  then  we  have  16  X  112  =  1792  Ibs. 
.==  the  force ;  wherefore, 

1792  X  53  224000 

r^T656^roT==-159F6  *  14°'  breadth  and  CUbe  °f 
depth. 


172  WHEELS. 

Now,  let  us  suppose  that  the  breadth  is  two  inches,  we 
Amst  divide  this  140  by  it,  whence, 

140 
-  =  70,  the  cube  of  the  depth, 

9 

ai.d  the  cube  root  of  70  will  be  found  =  4*121,  which  is 
the  depth  of  each  arm. 

When  the  depth  at  the  axis  is  intended  to  be  double  of 
the  depth  at  the  rim,  the  number  1640  is  to  be  used  in  the 
rule  instead  of  2't56. 

The  tables  vhich  follow  will  be  found  in  the  highest 
degree  useful  to  <»he  practical  mechanic. 


VHEKLS. 


173 


ABLE     OF     PITCHES    OF    WHEELS    IN    ACTUAL    USE     IN     MILL 
WORK. 


v  £.  £v.  £•  s.  5  a 

3     ffl     Ol     O>     (D     2     ™> 

?   "i  -»  •?  -i   9  9 


X 

B 


*•<    I    !     !    I    «    5   CD  a   tp  g> 

ca  :   :   :  :   :  P  -7-  -«-7 


fr 


Honet'  power. 


»-,-  l-i- 


Breadth  of  teeth  in  incho. 


_____,;  _    ,i    ,  :    ,i    ^: 

<iOCOOOOi         *>.  W  W  00  63  SO 


COCO 


Revolves  per 
minute. 


ODQOO'OlO9taW»^.COfcS*fcO»         r.   -   r. 

H-  H-"  H-  1—  > 

OO>-^-O500O<J<lO          6S          WOO 


-i          i^COiO  —   _^   ^.   tC   JC 

_          tOQDO          ODOO-^Oi 


Cl   W  62 

OOtS 


'    Teeth. 


CO 

o       cotocn       oopcpfcOH-apo  "*? 

to  co  H-  ^i  co 


Revolves  per 


tOf-  CCK/i  00(0 


Breadth  proportional  lo  10 
horses'  power,aiid  prticot 
velocity. 


"    «o  en  TO  c)  «       totoaoc 
QOW-^ltntt  en        en 


tionalto  10liors«'[iower,  I 

at  3  f.  p.  second,  that  is,  i 

reduciDealltheezamplei  ' 

to  the  same  deaom.  • 


15* 


174  WHKELS. 


EXPLANATION  OF  REFERENCES,  &c.,  IN  THE  FOREGO- 
ING  TABLE. 

1  The  only  defect  in  this  geering,  which  has  been  1 6  years  at  work, 
is  the  want  of  breadth  in  the  spur-wheel  and  pinion :  they  ought  to  have 
been  6  inches  or  more,  as  they  will  not  last  half  so  long  as  the  bevel- 
wheels  and  pinions  connected  with  them. 

3  Has  been  1 6  years  at  work.     The  teeth  are  much  worn. 

3  Has  been  16  years  at  work.     This  geering  is  found  rather  too  nar- 
row for  the  strain,  as  it  is  wearing  much  faster  than  the  rest  of  the  wheels 
in  the  same  mill. 

4  and  5  This  wheel  has  wooden  teeth,  and  has  been  working  for  three 
years. 

6  This  is  a  better  pitch  for  the  power  than  the  following. 

7  This  pitch  has  been  found  to  be  too  fine. 

In  the  foregoing  table  the  wheels  are  all  reduced  to  what 
may  be  called  one  denomination. — 1st.  By  proportioning 
all  their  breadths  to  what  they  should  be,  to  have  the  same 
strength,  if  the  resistance  were  equal  to  the  work  of  a  steam 
engine  of  ten  horses'  power.  2d.  By  supposing  their  pitch- 
lines  all  brought  to  the  same  velocity  of  three  feet  per  se- 
cond, and  proportioning  their  breadth  accordingly.  This 
particular  velocity  of  three  feet  per  second  has  been  chosen, 
because  it  is  the  velocity  very  common  for  overshot  wheels. 
Such  cases  as  appear  to  have  worn  too  rapidly,  are  marked, 
which  may  tend  to  discover  the  limit  in  point  of  breadth. 

TABLE    OF    PITCHES. 

THE  succeeding  table  of  pitches  of  wheels  was  drawn  up 
in  the  following  manner : — The  thickness  of  the  teeth  in 
each  of  the  lines  is  varied  one-tenth  of  an  inch.  The 
breadth  of  the  teeth  is  always  four  times  as  much  as  their 
thickness.  The  strength  of  the  teeth  is  ascertained  by 
multiplying  the  square  of  their  thickness  into  their  breadth, 
taken  in  inches  and  tenths,  &c.  The  pitch  is  found  by 
multiplying  the  thickness  of  the  teeth  by  2-1.  The  num- 
ber that  represents  the  strength  of  the  teeth,  will  also  repre- 
sent the  number  of  horses'  power,  at  a  velocityxof  about 
four  feet,  per  second.  Thus,  in  the  table  where  the  pitch 
is  3'15  inches,  the  thickness  of  the  teeth  1-5  inches,  and 
the  breadth  6  inches,  the  strength  is  valued  at  13|  horses 
power,  with  a  velocity  of  four  feet  per  second  at  the  pitch 
line. 


WHEELS. 


175 


A  Table  of  Pitches  of  Wheels,  ivith  the  breadth  and  thick 
ness  of  the  teeth,  and  the.  corresponding  number  of 
horses'  power,  moving  at  the  pitch-line  at  the  "ate  of 
three,  four,  six,  and  eight  feet,  per  second. 


Pitch  in 
inches. 

Thick- 
ness of 

inches. 

Briathh 
of  tef>th 
ill  inches. 

Strengih  of 

ll'rlhjiT  lll>. 

of  horses' 
]*  nvcr.  at  4 

cund. 

Horses' 
power  at  3 
feet  per 
second.  • 

Horses' 
po*er  at  6 
feet  per 
second. 

Horses' 
power  at  3 
feet  per 
second. 

3-99 

1-9 

7-6 

27-43 

20-57 

41-14 

54-85 

3-78 

1-8 

7-2 

23-32 

17-49 

34-98 

46-64 

3-57 

1-7 

6-8 

19-65 

14-73 

29-46 

39-28 

3-36 

1-6 

6-4 

16-38 

12-28 

24-56 

32-74 

3-15 

1-5 

6- 

13-5 

10-12 

20-24 

26-98 

2-94 

1-4 

5-6 

10-97 

8-22 

16-44 

21-92 

•2-73 

1-3 

5-2 

8-78 

6-58 

13-16 

17-34 

2-52 

1-2 

4-8 

6-91 

5-18 

10-36 

13-81 

2-31 

1-1 

4.4 

5-32 

399 

7-98 

10-64 

2-1 

1-0 

4- 

4-0 

3-0 

6-0 

8-0 

1-89 

•9 

3-6 

2-91 

2-18 

4-36 

5-81 

1-68 

•8 

3-2 

2-04 

1-53 

3-06 

3-08 

1-47 

•7 

2-8 

1-37 

1-027 

2-04 

2-72 

1-26 

•6 

2-4 

•86 

•64 

1-38 

1-84 

1-05 

•5 

2- 

•5 

•375 

•75 

1- 

HYDROSTATICS. 


HYDROSTATICS  comprehends  all  the  circumstances  of  the 
pressure  of  non-elastic  fluids,  as  water,  mercury,  <fcc.,  and 
of  the  weight  and  pressure  of  solids  in  them,  when  these 
fluids  are  at  rest.  Hydrodynamics,  on  the  other  hand,  refers 
to  the  like  circumstances  of  fluids  in  motion. 

The  particles  of  fluids  are  small  and  easily  moved  among 
themselves. 

Motion  or  pressure  in  a  fluid  is  not  in  one  straight  line 
in  the  direction  of  the  moving  force,  but  is  propagated  in 
every  direction,  upwards,  downwards,  sidewise,  and  oblique. 

From  this  property  it  is,  that  water  will  always  tend  to 
come  to  a  level,  for  if  two  cisterns  be  filled  with  water,  the 
one  10  feet  deep,  and  the  other  6,  there  will  be  more  pres- 
sure on  the  bottom  of  the  10  feet,  than  the  6  feet  cistern ; 
and,  if  the  bottoms  of  both  cisterns  be  on  a  level,  and  a 
pipe  be  made  to  communicate  between  them,  then  the  water 
in  the  deep  cistern  will  exert  a  greater  pressure  than  that 
in  the  other,  and  will  cause  the  other  to  rise  till  their  pres- 
sures become  equal,  that  is,  when  their  surfaces  come  to  a 
level ;  and  this  will  hold  true,  however  different  the  sur- 
faces of  the  two  cisterns  may  be  in  area.  Hence,  if  water 
be  communicated  through  pipes  between  any  number  of 
places,  it  will  rise  to  the  same  level  in  all  the  places,  whe- 
ther the  pipes  be  straight  or  bent,  wide  or  narrow  ;  and  any 
fluid  surface  will  rest  only  when  that  surface  is  level. 

If  a  vessel  contain  water,  the  pressure  on  any  point  in 
the  sides  or  bottom,  is  proportional  to  the  perpendicular 
height  of  the  fluid,  above  that  point,  in  the  side  or  bottom. 

The  pressure  of  a  fluid  upon  a  horizontal  base,  is  equal 
to  the  weight  of  a  column  of  the  fluid,  of  the  area  of  the 
base  multiplied  by  the  perpendicular  height  of  the  fluid, 
whatever  be  the  shape  of  the  containing  vessel :  so  that  by 
a  long  and  very  small  pipe,  the  strongest  casks  or  vessels 

176 


HYDROSTATICS.  177 

may  be  burst  asunder  by  the  pressure  of  a  very  small  quan- 
tity of  water. 

Ex. — Into  a  square  box  a  tube  is  fixed,  so  that  it  shall 
stand  perpendicularly ;  the  area  of  the  bottom  of  the  box 
is  9  square  feet,  and  the  height  of  the  top  of  the  tube  above 
the  bottom  of  the  box  is  5  feet,  and  therefore  the  pressure 
on  the  bottom  is  5  X  9  =  45  cubic  feet  of  water.  Now 
the  weight  of  one  cubic  foot  of  water  is  found  to  be  very 
nearly  1000  ounces  avoir.,  therefore,  45  X  1000  =  45,000 
ounces,  =  1  ton,  5  cwt.  0  qrs.  12  Ibs.  8  oz. 

The  content  in  imperial  gallons  of  any  rectangular  cis- 
tern may  be  found  thus, 

cistern's  content  in  cubic  feet  X  6-232, 
or  cistern's  content  in  inches  x  '003607, 
cistern's  content  in  cubic  inches 

277-274 
content  in  imperial  gallons. 

From  these  rules,  which  are  approximate,  it  is  easy  to 
see  that  of  the  three,  the  length,  breadth,  and  depth  of  a 
cistern,  any  two  being  given  the  third  may  be  found,  so 
that  the  yessel  shall  contain  any  given  number  of  gallons, 
thus, 

number  of  gallons 

— -=  the  third 
any  two  dimensions  in  feet  X  6-232 

dimension  in  feet. 

For  the  content  in  gallons  of  a  cylindric  vessel, 

diameter8  X  length  X  4-895, 

if  the  dimensions  are  in  feet,  but  if  the  diameter  be  in 
inches,  use  '034  instead  of  4-895,  and  should  both  dimen- 
sions be  in  inches,  use  -002832,  or  divide  by  352-0362. 
Also  when  the  length  and  diameter  are  in  feet, 
number  of  gallons 


length  x  4-895 
number  of  gallons 


=  diameter 
=  length. 


diameter2  x  4-895 

For  a  sphere  we  have  diameter3  X  3-263  =  content  in 
gallons,  the  diameter  being  in  feet,  but  when  the  diameter 
s  in  inches,  use  the  number  '001888.  These  rules  may  be 
illustrated  by  the  following  examples. 

The  length  of  a  cistern  being  8  feet,  its  breadth  4-5,  and 
depth  3,  then  will  its  content  be  8  X  4-5  X  3  =  108  cubic 


178  HYDROSTATICS. 

feet,  hence  108  X  6-232  =  673-06  gallons  may  be  con- 
tained in  it. 

It  is  required  that  a  cistern  should  contain  1000  gallons, 
but  must  not  exceed  10  feet  in  length  and  5  in  breadth, 
wherefore, 

'1000  1000 

10  x  5  x  6-232  ~  3TF6  = 

A  cylinder  is  6-5  feet  long  and  3  inches  diameter,  there- 
fore 6-5  X  32  X  -034  =  1-989  gallons  that  it  will  contain. 

A  pipe  is  to  be  made  20  inches  in  length,  what  must  be 
its  diameter  so  that  it  shall  contain  5  gallons  ? 

x  354 


20 


=  9*4  inches. 


The  quantity  of  pressure  upon  any  plane  surface  on  which 
a  fluid  rests,  is  equal  to  the  pressure  upon  the  same  plane 
placed  horizontally  at  the  depth  of  its  centre  of  gravity. 

If  any  plane  surface,  either  vertical  or  inclined,  be  placed 
in  a  fluid,  the  centre  of  pressure  of  the  fluid  on  the  plane 
is  at  the  centre  of  percussion,  the  surface  of  the  fluid  being 
supposed  the  centre  of  motion.  Thus  it  will  be  found  that 
in  a  cistern  whose  sides  are  vertical,  the  centre  of  pressure 
on  the  sides  is  two-thirds  from  the  top,  which  is  also  the 
centre  of  percussion. 

To  ascertain  the  whole  pressure  on  a  flood-gate,  or  other 
surface  exposed  to  the  pressure  of  water,  a  very  near  ap- 
proach to  the  truth  may  be  made  by  these  rules — the  breadth 
and  depth  being  taken  in  feet. 

31-25  X  breadth  X  depth3  =  pressure  in  Ibs. 

•2727  X  breadth  X  depth3  =  pressure  in  cwts. 

If  the  gate  be  wider  at  the  top  than  bottom, 

/breadth  at  top  —  breadth  at  bottom\ 
31-25  X  (-  -)   +  breadth 

\  9  f 

at  bottom  X  depth3  =  pressure  in  Ibs. ;  and  -2727,  used  in 
stead  of  31-25,  will  give  the  pressure  in  cwts.,  nearly 

Exam. — What  is  the  pressure  upon  a  rectangular  flood- 
gate, whose  breadth  is  25  feet,  and  depth  below  the  surface 
of  the  water  12  feet? 

31-25  X  25  X  12a  ==  112500  Ibs.  pressure. 

If  the  breadth*  at  top  be  28  feet,  that  at  bottom  22,  and 
the  height  12,  as  before,  then, 


HYDROSTATICS.  179 

oa 22 

31-25  X -  4-  22  X  12a  =  108000  Ibs.  pressure. 

3 

The  weight  of  a  cubic  foot  of  river  water  is  about  ^  of 
a  cwt.  The  pressure  at  the  depth  of  30  feet  is  about  13 
Ibs.  to  the  square  inch.  And  at  the  depth  of  36  feet  the 
pressure  is  about  1  ton  to  the  square  foot.  The  weight  of 
an  imperial  <rallon  of  water  is  about  10  Ibs. 

Ex. — What  is  the  pressure  at  the  depth  of  120  feet  on  a 
square  inch  ? 

30  :  120  :  :  13  :  52  =  the  pressure,  and  at  the  same 
depth,  36  :  120  :  :  1  :  3i  tons  on  the  square  foot. 

It  is  not  difficult  to  see  that  the  strength  of  the  vessels 
or  pipes  which  contain  or  convey  water  must  be  regulated 
according  to  the  pressure. 

The  thickness  of  pipes  to  convey  water  must  vary  in  pro- 
portion to  the  height  of  the  head  of  water  X  diameter  of 
pipe  -T-  the  cohesion  of  one  square  inch  of  the  material  of 
which  the  pipe  is  composed. 

By  experiment  it  has  been  found  that  a  cast  iron  pipe  15 
inches  diameter  and  |  of  an  inch  thick  of  metal,  will  be 
sufficiently  strong  for  a  head  600  feet  high.  A  pipe  of  oak 
15  inches  diameter  and  2  inches  thick,  is  sufficient  for  a 
head  of  180  feet.  When  the  material  is  the  same,  the 
thickness  of  the  material  varies  with  the  height  of  head  X 
diameter  of  pipe 

Ex. — What  must  be  the  thickness  of  a  cast  iron  pipe  10 
inches  diameter  for  a  head  of  360  feet  ? 
360  X  10  X  ! 

600~X~15~~  =  T*       a"  *        thlckness- 

If  the  same  pipe  is  to  be  made  of  oalc,  then 

300  x  10  X  2 

- — —  =  2f  thickness  in  inches. 

When  conduit  pipes  are  horizontal  and  made  of  lead, 
their  thicknesses  should  be  2|,  3,  4,  5,  6,  7,  8  lines,  when 
ihe  diameters  are  1,  1|.  2,  3,  4£,  6,  7  inches — and  when 
the  pipes  are  made  of  iron,  their  thickness  should  be  1,  2, 
3f  4,  5,  6,  7,  8  lines,  when  their  diameters  are  1,2,  4,  6, 
B,  10,  12. 

The  plumber  should  be  aware  that  the  tenacity  of  lead  is 
increased  four  times,  by  adding  1  part  of  zinc  to  8  of  lead. 

When  the  vessel  which  contains  the  water  has,  besides  the 


180  HYDROSTATICS. 

pressure  arising  from  the  weight  of  the  water,  to  resist  an 
additional  pressure  exerted  by  some  force  on  the  water,  as 
in  Bramah's  press,  where  the  pressure  exerted  by  means  of 
a  force  pump  on  the  water  in  a  small  tube,  which  commu- 
nicates with  a  large  cylinder,  is,  by  the  principles  stated 
before  in  this  chapter,  multiplied  on  the  piston  of  the 
cylinder  as  often  as  the  area  of  the  tube  is  contained  in  the 
area  of  the  piston  of  the  cylinder.  If  the  area  of  the  tube 
(  be  one  inch,  the  area  of  the  piston  92  inches,  and  if  the 
pressure  on  the  water  in  the  tube  be  16  Ibs.,  then  the  pres- 
sure on  the  piston  will  be  16  X  92  =  1472  Ibs. 

The   annexed   figure    and    description    taken    from    the 
Popular   Encyclopedia,  'will   give    a   clearer   idea   of  the 
operation  of  this  press.     "  Here  AB  is    the  bottom  of  a 
hollow  cylinder,  into  which  a  piston 
P  is  accurately  fitted.     Into  the  bot-       f  p 

torn  of  this  cylinder  there  is  intro- 
duced a  pipe  C  leading  from  the 
forcing  pump  D  ;  water  is  supplied 
to  this  pump  by  a  cistern  below,  from 
which  the  pipe  E  is  led,  being  fur-  D 
nished  with  a  valve  opening  upwards 
where  it  is  joined  to  the  pump  barrel.  E 
Where  the  pipe  C  enters  into  the  pump  barrel  there  is  alsi 
a  valve  opening  outwards  into  the  pipe ;  consequently, 
when  the  piston  D  rises,  this  valve  shuts,  and  the  valve  at 
ihe  cistern  pipe  opens,  and  the  fluid  rises  into  the  pump 
barrel.  The  top  of  the  piston  rod,  P,  is  fixed  in  the  bottom 
of  the  board  on  which  the  goods  are  laid,  and  when  the 
piston  rises  the  goods  are  pressed  against  the  top  of  the 
framing  of  the  machine.  When  the  piston  begins  to  de- 
scend, the  cistern  valve  shuts,  and  the  water  is  forced 
through  the  pipe  C  into  the  large  cylinder  AB  ;  and  by  the 
law  of  fluids  before  alluded  to,  whatever  pressure  be  exerted 
by  the  piston  D  on  the  surface  of  the  water  in  the  pump, 
will  be  repeated  on  the  piston  of  the  large  cylinder  AB  as 
many  times  as  the  area  of  the  small  piston  I)  is  contained 
in  the  area  of  the  large  piston  AB ;  that  is,  if  the  area  of 
the  pump-piston  were  one  square  inch,  and  that  of  the 
cylinder  100  inches,  and  if  the  piston  were  forced  down 
with  a  pressure  of  10  Ibs.,  then  the  whole  pressure  on  the 
bottom  of  the  piston  AB  will  be  10  times  100,  that  is,  1000 
Ibs.  When  the  page  which  is  now  before  the  reader  was  taken 


HYDROSTATICS.  181 

wet  off  the  types,  it  was  all  deeply  indented  in  consequence 
of  the  pressure  of  the  printing  press  ;  but  after  being  dried,  it 
was  subjected  to  the  action  of  Bramah's  press,  by  which 
process,  as  will  be  seeYi,  these  indentations  have  been  nearly 
obliterated.  In  the  press  by  which  this  has  been  accom- 
plished, the  pump  has  a  bore  of  three-fourths  of  an  inch  in 
diameter,  and  the  cylinder  one  of  eight  inches,  their  areas 
are  therefore  to  one  another,  as  9- 16th  to  64,  (the  squares 
of  the  diameters,)  that  is,  as  1  is  to  113;  hence  if  the 
pressure  upon  the  pump-cylinder  be  56  Ibs.,  (which  can  be 
easily  effected  by  boys,)  the  pressure  upon  the  piston  of 
the  large  cylinder  will  be  56  X  113,  that  is,  6'328  Ibs. 
This  astonishing  power  has  also  been  employed  in  the 
construction  of  cranes." 

To  ascertain  the  thickness  of  metal  necessary  for  the 
cylinder  of  such  presses,  this  rule  will  serve  : 

pressure  per  square  inch  X  radius  of  cylinder 
cohesion  of  the  metal  per  square  in.  —  pressure 
thickness  of  metal  necessary  for  the  cylinder  to  sustain  the 
pressure.     The  pressure  being  in  Ibs. 

NOTE. — The  cohesive  force  of  a  square  inch  of  cast  iron 
is  18,000  Ibs. 

What  is  the  thickness  of  metal  in  a  cast  iron  press  whose 
cylinder  is  12  inches  diameter,  the  pressure  being  1-5  tons 
on  the  circular  inch  ? 

A  circular  inch  is  to  a  square  inch  as  0'7854  to  1,  there- 
fore 1'5  tons  per  circular  inch  =  1'9  per  square  inch  = 
4256  Ibs. 

Here  we  have 


18000  —  4256 

What  is  the  thickness  of  metal  in  a  press  of  yellow  brass, 
whose  cylinder  is  10  inches  in  diameter,  and  which  is  in 
tended  for  a  pressure  of  2  tons  to  the  square  inch  ? 

The  cohesive  force  of  yellow  brass  being  17958,  we  have 
by  the  same  rule, 

2  tons  =  4480  Ibs. 

-  =  T66  inches,  the  thickness   of  the 


17958  —  4480 
metal. 


182  HYDROSTATICS. 

When  the  diameter  remains  the  same,  the  thickness  ap- 
pears to  increase  with  the  increase  of  pressure= 


FLOATING   BODIES. 

WHEN  any  body  is  immersed  in  water,  it  will,  if  it  be  of 
the  same  density  of  the  water,  remain  suspended  in  any 
place  ;  but  if  it  be  more  dense  than  the  water  it  will  sink, 
and  if  less  dense  it  will  float. 

Bodies  immersed  and  suspended  in  a  fluid  lose  the  weight 
of  an  equal  bulk  of  the  fluid,  and  the  fluid  acquires  the 
weight  that  the  body  loses :  also,  bodies  floating  on  a  fluid 
lose  weight  in  proportion  to  the  quantity  of  fluid  they  dis- 
place. 

When  a  body  floats  upon  the  water,  it  will  sink  in  the 
water  till  the  water  which  is  displaced  be  equal  in  weight 
to  the  weight  of  the  body. 

When,  a  body  floats  on  a  fluid,  it  will  only  be  at  rest 
when  the  centre  of  gravity  of  the  body  and  the  cen're  of 
gravity  of  the  displaced  fluid  are  in  the  same  vertical  line ; 
and  the  lower  the  centre  of  gravity  is,  the  more  stable  will 
the  body  be. 

The  buoyancy  of  casks,  or  the  load  which  they  will  carry 
without  sinking,  may  be  estimated  at  about  10  Ibs.  to  the 
ale  gallon,  or  282  cubic  inches  of  the  content  of  the  cask. 


SPECIFIC  GRAVITY. 

SPECIFIC  gravity  is  the  relative  weight  of  any  body  of  a 
certain  bulk,  compared  with  the  weight  of  some  body  taken 
as  a  standard  of  the  same  bulk.  The  standard  of  compari- 
son is  water ;  one  cubic  foot  of  which  is  found  to  weigh 
1000  ounces  avoir,  at  a  temperature  of  00°  of  Fahrenheit, 
so  that  the  weight  expressed  in  ounces  of  a  cubic  foot  of  any, 
body,  will  be  its  specific  gravity,  that  of  water  being  1000. 

To  determine  the  specific  gravity. 

If  a  body  be  a  solid  heavier  thaft  water — Weigh  it  first 
carefully  in  air,  and  note  this  weight;  then  immerse  it  in 
water,  and  in  this  state  note  its  weight.  Then  divide  the 
body's  weight  in  air  by  the  difference  of  the  weights  in  air 
and  water,  the  quotient  is  the  specific  gravity. 


SPECIFIC    GRAVITY. 


183 


If  a  body  be  a  solid  lighter  than  u'atcr — Tie  a  piece  of 
metal  to  it,  so  that  the  compound  may  sink  in  water — then 
to  the  weight  of  the  solid  itself  in  air,  add  the  weight  of  the 
metal  in  water,  and  from  this  sum  subtract  the  weight  of  the 
compound  in  water,  which  difference  makes  a  divisor  to  a 
dividend,  which  is  the  weight  of  the  solid  in  air,  then  the 
quotient  will  he  the  specific  gravity. 

If  (lie  body  be  a  Jlvid — Take  a  solid,  whose  specific 
gravity  is  known,  and  that  will  sink  in  the  fluid  ;  then  take 
the  difference  of  the  weights  of  the  solid  in  and  out  of  the 
fluid,  and  multiply  this  difference  by  the  specific  gravity 
of  the  solid ;  then,  this  product  divided  by  the  weight  of 
the  body  in  air,  will  give  the  specific  gravity  of  the  fluid. 

On  these  principles  there  has  been  constructed  tables  of 
specific  gravities,  one  of  which  we  insert.  The  column, 
specific  gravity,  may  be  taken  to  represent  the  weight  cf 
a  cubic  foot. 


TABLE    OF   SPECIFIC   GRAVITIES. 


METALS. 


Specific  Gravity, 

Arsenic, 5763 

Cast  antimony, 6702 

Cast  zinc, 7190 

Cast  iron, 7207 

Cast  tin, 7291 

Bar  iron 7788 

Cast  nickel, 7807 

Cast  cobalt, 7811 

Hard  steel, 7816 

Soft  steel, 7833 

Cast  brass, 8395 

Cast  copper 8788 


Specific  Gravity 

Cast  bismuth, 9822 

Cast  silver, 10474 

Hammered  silver,  ...  10510 

Cast  lead, 11352 

Mercury, 13568 

Jewellers'  gold, 15709 

Gold  coin, 17647 

Cast  gold,  pure, 19258 

Pure  gold,  hammered,  19361 

Platinum,  pure, 19500 

Platinum,  hammered,  20336 
Platinum  wire, 21041 


STONES,    EARTHS,    ETC. 


Brick 2000 

Sulphur, 2033 

Stone,  paving, 2416 

Stone,  common, 2520 


Pebble, 2664 

Slate 2672 

Marble, 2742 

Chalk 2784 


Granite,  red, 2654  Basalt, 2864 

Glfess,  green, 2642  Hone,  white  razor,  ...  2876 

Glass,  white, 2892  Limestone, 3179 

Glass,  bottle, 2733 


184 


HYDROSTATICS. 


RESINS,  ETC. 

Specific  Gravity. 


Specific  Gr»Ti*y 


Wax, 897 

Tallow, :....  945 


Bone  of  an  ox, 1659 

Ivory, 1822 


LIQUIDS. 


Air  at  the  earth's   sur- 
face,   


If 


Oil  of  turpentine, 870 

Olive  oil, 915 


Distilled  water, 1000 

Sea  water, 1028 

Nitric  acid, 1218 

Vitriol..,  .  1841 


WOODS. 


Cork, 246 

Poplar, 383 

Larch, 544 

Elm  and  new  English  fir,556 
Mahogany,  Honduras,- -560 

Willow, 585 

Cedar, 596 

Pitch  pine,  560 

Pear  tree, 661 

Walnut, 671 

Fir,  forest, 694 

Elder, 695 

Beech, 696 

Cherry  tree, 715 

Teak, 745 


Maple  and  Riga  fir, 750 

Ash  and  Dantzic  oak,  "760 

Yew,  Dutch, 788 

Apple  tree, 793 

Alder, 800 

Yew,  Spanish, 807 

Mahogany,  Spanish, 852 

Oak,  American, 872 

Boxwood,  French, 912 

Logwood, 913 

Oak,  English, 970 

Do.  sixty  years  cut,- -1170 

Ebony, 1331 

Lignum  vitae, 1333 


Specific  gravity  of  gases,  that  of  atmospheric  air  being 
=  1. 


Hydrogen, 0-0694 

Carbon, 0-4166 

Steam  of  water, 0-481 

Ammonia, 0-5902 

Carburetted  hydrog., 0-9722 

Azote, 0-9723 

Oxygen, 1-1111 

Muriatic  acid, 1-2840 


Carbonic  acid, 1'5277 

Alcohol  vapour, 1-6133 

Chlorine,  2-500 

Nitrous  acid,  2-638 

Sulphuric  acid, 2-777 

Nitric  acid, 3-75 

Oil  of  turpentine, ••••5-013 


NOTE. — The  specific  gravity  of  atmospheric  air  at  a  tem- 
perature of  60°  Fah.  and  barometric  column  30  inches  is 
1-22  according  to  M.  Arago,  and  in  round  numbers  we  may 
regard  water  as  825  times  heavier  than  air. 


SPECIFIC    GRAVITY.  185 

The  preceding  table  will  be  found  of  the  utmost  use  in 
determining  the  weight  and  magnitude  of  bodies. 
To  find  the  magnitude  of  a  body  from  its  weight  : 

weight  of  body  in  ounces 

.  —    —  r^  —  —  =  content  in  cubic  feet. 

its  specific  grav.  in  table 

How   many  cubic  feet  are  in  one  ton  of  mahogany  ? 
Here  20  x  112  x  16  =  35840  ounces  in  a  ton  ;  therefore. 

35840 

—  =  64  cubic  feet. 
oou 

Had  the  timber  been  fir,  then 

—-—=  64-46  cubic  feet. 

ODD 

Or  English  oak  : 

35840 

=  36-94  cubic  feet. 

«7/  U 

To  find  the  weight  of  a  body  from  its  bulk  : 

cubic  feet  x  specific  gravity  =  weight  in  ounces. 
What  is  the  weight  of  a  log  of  larch,  14  feet  long,  2£ 
broad,  and  1$  thick? 

Here  2-5  X  1'25  X  14  =  43-75;  then, 
43-75  x  544  =  23800  ounces  =  13  cwt.  1  qr.  3  Ibs.  8  oz. 
What  is  the  weight  of  a  cast  iron  ball,  2  inches  diameter? 
Here  the  content  of  the  globe  will  be  2s  x  -5236  =  4-1888 
cubic  inches   =  -00242  feet,  and  then  -00242  X  7271  = 
17-29  ounces  =  1-08  Ibs. 

A  bullet  of  lead  of  the  same  magnitude  would  be  -00242 
X  11344  =  27-44  ounces  =  1-71  Ibs. 

If  we  wish  to  determine  the  quantity  of  two  ingredients 
in  a  compound  which  they  form, 

Let  H  be  the  weight  of  the  heavy  body. 
A,  its  specific  gravity. 
L,  the  weight  of  the  lighter  body. 
/,  its  specific  gravity. 
C,  the  weight  of  the  compound. 
c,  its  specific  gravity. 
Then, 

(c  —  /)  x  h 

W  -  f.  --  x  C  =  H. 

(h  —  /)  x  c 

also, 

- 


16* 


186  HYDI50.STAT10S.     - 

Ex. — A  mixture  of  gold  and  silver  weighed  170  Ibs.  and 
its  specific  gravity  was  15030  ;  hence 

h  (by  the  table)  ==  19326.  /  ==  10744 
c  =  15630  C  =  170  Ibs.  wherefore,  by  the  rule, 

(19326—  15630)  x  10744  _  39709824 

(1932ft— 10744)  X  15630  *  134136660 X 

=  -296  X  170  =  50-32  Ibs.  of  gold  ; 

consequently  there  will  be  170  —  50-32  =  119-68  Ibs.  of 

silver. 

The  weight  of  bodies — their  magnitudes  and  also  their 
quantities  in  a  compound,  may  thus  be  found  by  means  of 
a  table  of  specific  gravities  ;  and  for  the  more  expeditious 
calculation  in  practice  we  add  the  following  memoranda: 

430-25  cubic  inches  of  cast  iron  weigh  one  cwt.,  as  also 
397-60  of  bar  iron,  368-88  of  cast  brass,  352-41  of  cast 
copper,  and  372-8  of  cast  lead. 

14-835  cubic  feet  of  common  paving  stone  weigh  one  ton, 
as  also  14-222  of  common  stone,  13-505  of  granite,  13-070 
of  marble,  64-46  of  elm,  64  of  Honduras  mahogany,  51-65 
of  fir,  51-494  of  beech,  42-066  of  Spanish  mahogany,  and 
36-205  of  English  oak. 

For  wrought  iron  square  bars,  allow  100  inches  in  length 
of  an  inch  square  to  a  quarter  of  a  cwt. 

A  similar  cast  iron  bar  would  require  9  feet  in  length  for 
a  quarter  of  a  cwt.  One  foot  in  length  of  an  inch  square 
bar  weighs  3-|-  Ibs.  also  the  breadth  and  thickness  being 
taken  in,  £th  of  an  inch,  and  the  length  in  feet. 

length  x  breadth  X  thickness  X  7 

= — — =  the  weigh! 

in  avoirdupois  pounds. 

Ex. — An  iron  bar  10  feet  long,  3  inches  broad,  and  2j 
thick.  Here  3  inches  =  24,  and  2 5  =  20-8ths  ;  therefore, 

10  X  24  x  20  x  7  . 

—— =  233  Ibs. 

144 

For  the  weight  of  a  cast  iron  pipe  : 

The  length  being  taken  in  feet,  the  diameter  and  thick 
ness  of  metal  in  inches,  then  we  have 

0-0876  X  length  x  thickness  x  (inner  diameter  4 
thickness)  =  the  weight  in  cwts. 

For  a  leaden  pipe  the  rule  is, 

0-t382  X  length  X  thickness  x  (inner  diameter  -f 
thickness)  =  the  weight  in  cwts. 


SPECIFIC    GRAVITY.  187 

NOTE. — The  weight  of  a  cast  iron  pipe  is  to  a  leaden 
pipe  of  the  same  dimensions  nearly  as  7  is  to  11. 

Ex. — If  the  inner  diameter  or  bore  of  a  cast  iron  pipe 
be  3  inches,  and  its  thickness  $  of  an  inch ;  what  is  the 
weight  of  14  feet  of  it? 

•0876  X  14  x  $  x  (3  +  £)  =  -99645  cwt.  =  3  qrs. 
27  Ibs.  9  oz. 

A  leaden  pipe  is  12  feet  long,  the  bore  is  4  inches,  and 
thickness  of  metal  |  of  an  inch ;  therefore, 
•1382  X  12  X  *  X  (4  +  4-)  =  1-762  cwt.  =  1  cwt.  3  qrs.  1  Ib. 

For  the  weight  of  the  rim  of  a  fly-wheel.  Let  D  be  the 
diameter  of  the  fly,  exclusive  of  the  rim,  taken  in  inches ; 
then  take  the  difference  of  this  and  the  diameter  of  the  fly, 
including  the  rim,  and  call  this  difference  d,  T  being  the 
thickness  of  the  rim  of  the  fly,  from  side  to  side,  then  W3 
have 
•0073  x  T  x  d  x  (D  +  cT)  =  the  weight  of  the  rim  in  cwts, 

Ex. — If  the  interior  diameter  of  the  fly  be  100  inches 
=  D,  half  the  difference  of  the  exterior  and  interior  dia- 
meter 5  =  c/,  hence  if  the  rim  is  10  inches  broad,  as  the 
exterior  diameter  will  then  be  110,  and  let  the  thickness 
of  the  rim  be  4  inches  =  T,  then, 

•0073    <  4  X  5  X  (100  +  5)  =  15-33  cwts. 


188 


HYDROSTATICS. 


TABLE  A. 

Of  the  weight  of  1  lineal  foot  of  Swedish  iron,  of  all  breadths  and 
thicknesses,  from  1  tenth  of  an  inch  to  1  inch,  in  pounds  and  deci' 
•mal  parts. 


•1 

•2 

•3 

•4 

•5 

•6 

•7 

•8 
•270 
•541 

•9 

1-0 

lOths  of 
inches. 

•034 

•068 

•101 

•135 

•169 

•203 

•237 

•304 

•338 

•1 

•135 

•203 

•270 

•338 

•406 

•473 

•608 

•676 

•2 

•304 

•406 

•507 

•609 

•710 

•811 

•913 

1-014 

•3 

•541 

•676 

•811 

•947 

1-082 

1-217 

1-352 

•4 

•845 

1-014 

1-183 

1-352 

1-521 

1-690 

•5 

1-217 

1-420 

1-623 

1-826 

2-029 

•6 

1-657 

1-893 

2-130 

2-367 

•7 

2-164 

2-434 

2-657 

•8 

2-739 

3-043 

•9 

3-381 

1-0 

TABLE  B. 

Of  the  weight  of  1  lineal  foot  of  Swedish  iron,  of  all  breadths  and 
thicknesses,  from  1  inch  to  6  inches,  in  pounds  and  decimal  parts. 


1   u  14  1  11 

2 

«1 

3 

3* 

4 

5 

6 

in. 

3-38  4-23  5-07    5-91 

6-76 

8-45 

10-14 

11-83 

13-52 

16-91  20-29 

1 

5-29  6-34,   7-40 

8-45 

10-56 

12-68 

14-79  16-91 

21-13  25-36 

1* 

7-60    8-87 

10-14J12-67 

15-21 

17-75  20-29 

25-36 

30-43 
35Td 

ii 
i* 

(10-35 

11-83 
13-52 

14-78 

17-75 

20-71 

23-67 

29-58 

i 

16-91 

20-29 

23-67 

27-05 

33-81 

40-51 

2 

21-13 

25-36 

29-58 
35-50 

33-81 

42-26 

50-72 

2* 

30-43 

40-57 

50-72 

60-86 

3 

41-42 

47-34 

59-1  6  [71-00 

3* 

54-10 

67-62 

81-14 

4 
5 
T 

84-52 

101-44 

121-72 

SPECIFIC    GRAVITY. 


189 


TABLE  C. 


Of  the  weight  of  1  superficial  foot  of  Swedish  iron  plate  from  IQbth 
part  of  an  inch  thick  to  one  inch. 


Thickness  in 
parts  of  an  inch. 

Weight  in  Ibs. 

Thickness  in 

parts  of  a  a  inch. 

Weight  in  Ibs. 

•01 

•406 

•1 

4-057 

•02 

•811 

•2 

8-114 

•03 

1-217 

•3 

12-172 

•04 

1-623 

•4 

16-232 

•05 

2-029 

•5 

20-286 

•06 

2-434 

•6 

24-344 

•07   .. 

2-840 

•7 

28-401 

•08 

3-246 

•8 

32-458 

•09 

3-651 

•9 

36-516 

•10 

4-057 

1- 

40-573 

TABLE  D. 

Of  Multipliers  for  the  other  Metals,  whereby  their  weights  may  bf 
found  from  the  above  Tables. 


Metals. 

Multi- 
pliers. 

Metals. 

Multi- 
pliers. 

Platinum,  laminated 

2-846 
2-503 
2-486 
2-47 
1-457 
1-350 
1-344 
1-136 
1-132 

Copper,  cast  

1-128 
1-096 
1-080 
1-003 
1- 
•980 
•925 
•960 
•937 

Brass  wire 

Pure  gold,  hammered 

Steel  

Iron,  Swedish  

Pure  silver,  hammered 
-—  cast        »  i 

Pewter  

,  hammered  .  . 

Tin,  cast  

190 


HYDROSTATICS. 


TABLE  E. 

Table  of  the  weight  of  one  square  foot  of  different  rt.etals  in  various 
thicknesses,  in  pounds  and  decimal  parts. 


Thickness 
in  16tlis  of 
»ninch. 

Mai.  Iron. 
Swed. 

Mai.  Iron, 
English. 

Cast  Iron. 

Copper. 

Bns, 

Lead. 

1 

2-535 

2-486 

2-345 

2-860 

2-738 

3-693 

2 

5-070 

4-972 

4-690 

5-720 

5-476 

7-386 

3 

7-605 

7-458 

7-035 

8-580 

8-214 

11-079 

4 

10-140 

9-944 

9-380 

11-440 

10-952 

14-772 

5 

12-675 

12-130 

11-725 

14-300 

13-690 

18-465 

6 

15-216 

14-916 

14-670 

17-160 

16-428 

22-158 

7 

17-851 

17-402 

16-415 

20-020 

19-166 

25-851 

8 

20-280 

19-888 

18-760 

22-880 

2t;904 

29-544 

9 

22-815 

22-774 

21-105 

25-740 

24-642 

33-237 

10 

25-350 

24-260 

23-450 

28-600 

27-380 

36-930 

11 

27-885 

26-746 

25-795 

31-460 

30-118 

40-623 

12 

30-410 

29-232 

28-140 

34-320 

32-856 

44-316 

13 

32-945 

31-718 

30-485 

37-180 

35-594 

48-009 

14 

35-480 

34-204 

32-880 

40-040 

38-332 

51-702 

15 

38-015 

36-690 

35-225 

42-900 

41-170 

55-405 

16 

40-550 

39-176 

37-570 

45-760 

43-908 

59-098 

TABLE  F. 

Tabh  of  the  weight  of  1  foot  in  length  ofmal/eable  Iron  rod,  from 
one-fourth  to  6  inches  diameter. 


Diam. 

Weight. 

Diam. 

Weight. 

Diam. 

Weight. 

Diam. 

Weisht. 

Inch. 

)bs. 

Inch. 

Ibs. 

Inch. 

Ibs. 

Inch. 

Ibs. 

* 

•163 

U 

8-01 

31 

27-65 

41 

59-06 

1 

•368 

U 

9-2 

3| 

29-82 

41 

62-21 

5 

•654 

2 

10-47 

ai 

32-07 

5 

65-45 

1 

1-02 

2§ 

11-82 

31 

34-4 

51 

68-76 

n 
J 

1-47 

2| 

13-25 

31 

36-81 

51 

72-16 

i 

2 

2| 

14-76 

3i 

39-31 

51 

75-63 

1 

2-61 

2.1 

16-36 

4 

41-89 

5^ 

79-19 

i* 

3-31 

2| 

18-03 

4* 

44-54 

5S 

82-83 

H 

4-09 

21 

19-79 

4* 

47-28 

51 

86-56 

if 

4-94 

21 

21-63 

4| 

50-11 

51 

90-36 

ii 

5-89 

3 

23-56 

4£ 

53-01 

6 

94-25 

u 

6-91 

3£ 

25-56 

4j 

56 

SPECIFIC    GRAVITF. 


.91 


TABLE  G. 

Jbble  of  int.  weight  of  cast  iron  Pipes,  1  foot  long,  und  of  different 

thicknesses. 


Diam.  of 
bore. 

i 

Inch. 

i 

Inch. 

Inch. 

Inch. 

1 

Inch. 

I 

Inch. 

1 

Inch.   ' 

Inch. 

Ib 

Ib. 

Ib. 

Ib. 

Ib. 

Ib. 

Ib. 

1 

3-06 

5-06 

7-36 

9-97 

12-89 

16-11 

19-63 

U 

3-68 

5-98 

8*-  59 

11-51 

14-73 

18-25 

22-09  ' 

4-29 

6-9 

9-82 

13-04 

16-56 

20-4 

24-54 

H 

4-91 

7-83 

11-05 

14-57 

18-41 

22-55 

27- 

2 

5-53 

8-75 

12-27 

16-11 

20-25 

24-7 

29-45 

21 

6-14 

9-66 

13-5 

17-64 

22-09 

26-84 

31-85 

Si 

6-74 

10-58 

14-72 

19-17 

23-92 

28-93 

34-36 

2.? 

7-36 

11-5 

15-95 

20-7 

25-71 

31-14 

36-81 

3 

7-98 

12-43 

17-18 

22-19 

27-62 

33-29 

39-28 

31 

8-59 

13-34 

18-35 

23-78 

29-45 

35-44 

41-72 

3d 

9-2 

14-21 

19-64 

25-31 

31-3 

37-58 

44-18 

31 

9-76 

15-19 

20-86 

26-85 

33-13 

39-73 

46-63 

4 

10-44 

16-11 

22-1 

28-38 

34-98 

41-88 

49-1 

41 

11-1 

17-08 

23-37 

29-97 

36-87 

44-08 

51-6 

4£ 

11-66 

17-94 

24-54 

3  1  -4  4 

38-65 

46-17 

54- 

4! 

12-27 

18-87 

25-77 

32-98 

40-5 

48-32 

56-45 

5 

12-80 

19-78 

26-99 

34-51 

42-33 

50-46 

59- 

51 

13-5 

20-71 

28-23 

36-05 

44-18 

52-62 

61-36 

5i 

14-11 

21-63 

29-45 

37-58 

46-02 

54-76 

63-81 

C  3 

5? 

14-73 

22-5,5 

30-68 

39-12 

47-86  56-91 

66-27 

6 

15-34 

23-47 

31-91 

40-65 

49-7 

59-06 

68-73 

61 

15-95 

24-39 

33-13 

42-18 

51-54 

61-21 

72- 

61 

16-57 

25-31 

34-36 

43-72 

53-39 

63-36 

73-41 

6f 

17-18 

26-23 

:<r>-r>!) 

45-26   55-23 

65-28 

76-1 

7 

17-79 

27-15 

36-82 

46-79 

56-84 

67-65 

78-53  ; 

71 

18-41 

28-08 

38-05 

48-1 

58-91 

69-79 

81-      ! 

71 

19-03 

29- 

39-05 

49-86   60-74 

71-95 

83-45 

71 

19-64 

29-69 

40-5 

51-38    G2-r>9 

74-09 

86-     : 

8 

20-02 

30-83 

41-71 

52-92    61-42 

76-23 

88-35  ; 

.   'Q  > 
.  S4 

20-86 

31-74 

42-95 

54-45 

'66-26 

78-38   90-81 

8.| 

21-69 

32-9 

44-4 

56-21 

68-33 

80-76   93-49 

*  8| 

22-09 

33-59 

45-4 

57-52 

69-95 

82-68   95-72 

ft 

22-71 

34-52 

46-64 

59-07 

71-8 

84-84 

98-18' 

192 


HYDROSTATICS. 


Diini. 
of  bore. 

iL. 

Inch. 

Inch. 

Inch. 

Inch. 

f 

Inch. 

Inch. 

Inch. 

lb. 

lb. 

lb. 

lh. 

11). 

lb. 

lb. 

8i 

23-31 

35-43 

47-86 

60-59 

73-63 

86-97 

100-63 

9* 

23-93 

36-36 

49-09 

62-13 

75-47 

89-13 

103-1 

9| 

24-55 

37-28 

50-32 

(j:Mi6 

77-32 

91-28 

105-54 

10 

25-16 

38-2 

51-54 

65-2 

79-16 

93-42 

108 

10i 

25-77 

39-11 

52-77 

66-73 

80-99 

95-57 

110-44 

10* 

26-38 

40-04 

54 

68-26 

82-84 

97-71 

113 

10| 

27 

40-96 

55-22 

69-8 

84-67 

99-86 

115-35 

11 

27-62 

41-88 

56-46 

71-33 

b6-,";2 

102-01 

117-81 

Hi 

38-22 

42-8 

57-67 

72-86 

88-35 

104-15 

120-26 

"i 

28-84 

43-71 

58-9 

74-39 

90-19 

106-3 

122-71 

11* 

29-45 

44-64 

60-13 

75-93 

92-04 

108-45 

125-18 

12 

30-06 

45-55 

61-35 

77-46 

93-6 

110-6 

127-6 

Diam. 
of  bore. 

Inch. 

Inch. 

Inch. 

1 

Inch. 

H 

Inch. 

U 

Inch. 

li 

Inch. 

1* 

Inch. 

2 

Inch. 

Inch. 

lb. 

lb. 

lb. 

lb. 

lb. 

lb. 

lb.  " 

lb. 

lb. 

12* 

63-5 

97-3 

114 

132 

149 

167 

205 

243 

285 

13 

66 

101 

118 

137 

154 

173-5 

212 

252 

294 

18* 

68-4 

104-8 

122 

141-5 

160 

179 

219 

260 

304 

14 

75 

108-2 

126 

146 

165 

185 

227 

269 

314 

'*! 

73-4 

112-3 

130 

151 

170 

192 

234 

277 

324 

15 

75-8 

115-7 

135 

156 

176 

198 

242 

286 

334 

15* 

78-1 

119 

139 

.161 

181 

204 

250 

295 

344 

16 

80-7 

123 

143 

166 

187 

211 

257 

303 

355 

16* 

83-1 

126-5 

147 

170-1 

192 

217 

264 

312 

363 

17 

85-5 

130 

152 

178-5 

198 

223 

271 

322 

376 

17* 

87-8 

133-5 

157 

180-5 

203 

229 

278 

330 

383 

18 

90-5 

137 

161 

185 

209 

235 

285 

338  ' 

393 

18* 

93 

140-5 

166 

190 

217 

241 

293 

347 

402 

19 

95-5 

144-8 

169 

195 

222 

247 

300 

354 

412 

19* 

97-8 

148-5 

174 

200 

227 

253 

307 

363 

422 

20 

100 

152 

178 

205 

233 

259 

315 

372 

432 

|  20* 

102-5 

156   ':  183 

210 

238 

26 

323 

381 

442 

The  following  TABLE  of  the  weight  of  different  sub- 
stances used  in  building  and  engineering  requires  no  ex 
planation. 


SPECIFIC    GRAVITY. 


193 


NamMofBodim. 

Weight  of  a 
ubic  foot    in 
ounces. 

VfeifM    of    a 
cubic    foot    in 

p'.Ulllt. 

Wright  of  a 

cubic  inch  in 
ounces. 

Weight  of  a 
cubic    inch    in 

(•ouil'l-. 

Nuabcr  oT  I 
cubic  inche*  uu 
,  a  pound.  71 

Copper,  cast.... 
Copper,  sheet  •  • 

8788 
8915 
8396 
7271 
7631 
11344 
7833 
7816 
7190 
7292 
9880 
8784 
1520 
1250 
2000 
2416 
2672 
2742 
3160 
2880 
945 
240 
544 
556 
660 
696 
745 
760 
852 
970 
870 
915 
932 
927 
1000 
1028 
1015 
1026 
J13568 

549-25 
557-18 
52  -1-7  5 
151-43 
47(>-i)3 
709-00 
489-56 
488-50 
449-37 
455-75 
619-50 
649-00 
95-00 
78-12 
125-00 
151-00 
167-00 
171-37 
197-50 
180-00 
59-00 
15-00 
34-00 
34-75 
41-25 
43-50 
46-56 
47-50 
53-25 
60-62 
54-37 
57-18 
58-25 
57-93 
62-50 
64-25 
63-43 
64-12 
848-00 

5-086 
5-159 
4-852 
4-203 
4-410 
6-456 
1-527 
4-517 
4-156 
4-215 
5-710 
5-0775 
•8787 
•7225 
1-156 
1-396 
1-544 
1-585 
1-826 

i-r.fU 

•5462 
•138 
•315 
•321 
•382 
•403 
•431 
•440 
•493 
•561 
•503 
•529 
•539 
•536 
•578 
•594 
•587 
•593 
7-851 

•3178 
•3225 
•3037 
•263 
•276 
•4103 
•2833 
•2827 
•26 
•2636 
•3585 
•3177 
•055 
•0452 
•0723 
•0873 
•0967 
•0991 
•1143 
•1042 
•0342 
•0087 
•0197 
•0201 
•024 
•0252 
•027 
•0275 
•0308 
•0351 
•0315 
•0331 
•0337 
•03352 
•03617 
•0372 
•0367 
•037 
•4908 

3-146 
3-103 
3-293 

3-802 
3-623 
2-437 
3-530 
3-537 
3-845 
3-790 
2-789 
3-147 
18-190 
22-120 
13-824 
11-443 
10-347 
10-083 
8-750 
9-600 
29-258) 
115-200! 
50-823 
49-726 
41-890 
39-724 
37-113 
36-370 
32-449 
28-505 
31-771 
30-220 
29-665 
29-288 
27-648 
26-894, 
27-24$ 
26-949 
2-037 

Steel,  soft  

Coal  

Stone,  paving  •  • 

\Tirh1p.  .. 

rjiocc    . 

Pnrk   . 

F,lm  

Pine,  pitch  

Teak  

A«Vi  .  . 

Mahogany   
Oak    . 

Oil  of  turpentin 

Linseed  oil    ... 
Spirits,  proof-- 
-  Water,  distilled 

Tir  .  . 

17 


194  HYDROSTATICS. 

The  foregoing  tables  and  rules  will  be  found  of  the  ut- 
most service,  in  the  ready  calculation  of  the  weight  of 
materials  commonly  used  in  engineering. 

What  is  the  weight  of  a  bar  of  Swedish  iron  16.  feet  long, 
3  inches  broad,  and  !•!  inch  thick? 

By  table  B,  3-38  is  the  weight  of  a  piece  of  Swedish 
iron,  of  one  foot  long  and  one  inch  square,  wherefore, 

3-38  x  16  X  3  =  162-24;  and  then  for  the  fraction  -1, 
in  table  A,  we  have  for  the  weight  of  1  foot  by  •!  of  an 
inch  square  =  -034;  hence,  -034  x  3  x  16  =  16-32; 
wherefore  the  sum  of  the  two  =  162-24  +  16-32  =  178-56 
Ibs.,  the  weight/ 

If  we  wish  the  weight  of  an  equal  bar  of  cast  iron,  we 
must  employ  the  multipliers  in  table  D  ;  hence, 
178-56  X  '925  =  165-168. 

If  we  wished  it  for  lead,  the  multiplier  from  the  same 
table  being  1-457,  we  have, 

178-56  x  1-457  =  260-1619  Ibs.,  &c.,  &c, 

Then  if  lead  were  1  penny  per  pound,  the  price  of  such 
a  bar  would  be 


The  following  practical  rules  are  often  useful  and  may 
be  easily  remembered. 

For  round  bars  of  iron, 

diameter  (m)3   X  length   in  ft.    X  2-6  =  weight  of 
wrought  iron  in  Ibs. 

diameter  (m)a   X  length  in  ft.  x  2-48   =  weight  of 
cast  iron  bars  in  Ibs. 

A  cylindrical  bar  is  2  inches  diameter  and  29  inches  long, 
therefore,  2s  x  2-5  x  2-6  =  26  Ibs.  if  it  be  wrought  iron, 
but  if  cast,  2s  x  2-5  x  2-48  =  24-8  Ibs. 

Multiply  the  sum  of  the  exterior  and  interior  diameters 
of  a  cast  iron  ring  by  the  breadth  and  thickness  of  the  rim, 
and  also  by  0  0074,  he  results  will  be  the  weight  in  cvvts. 


HYDRODYNAMICS. 


As  hydrostatics  embraces  the  consideration  of  fluids  at 
rest,  so  hydrodynamics  or  hydraulics  comprehends  the  cir- 
cumstances of  fluids  in  motion.  Of  this  science,  little, 
comparatively  speaking,  is  yet  known  ;  but  as  it  is  of  the 
utmost  importance  to  man,  we  will  endeavour  to  lay  before 
our  readers  a  statement  of  the  more  important  results  of 
recent  inquiry  into  it. 

If  a  fluid  move  through  a  pipe,  canal,  or  river,  of  various 
breadths,  always  filling  it,  the  velocity  of  the  fluid  at  dif- 
ferent parts  will  be  inversely  as  the  transverse  sections  of 
these  parts. 


Thus  let  there  be  a  canal,  AB,  of  various  breadths  at 
different  places,  then  will  the  velocity  in  the  portion  ab  be 
to  that  of  the  velocity  in  erf,  as  the  area  of  the  cross  section 
at  cd  is  to  that  at  ab,  and  the  velocity  at  ef  will  be  to  that 
at  cd  as  the  area  at  cd  is  to  the  area  at  ef,  being  always  in 
inverse  proportion. 

Suppose  the  velocity  at  ab  10  feet  per  second,  and  the 
area  there  100  feet,  then  if  the  area  at  cd  be  25  feet,  we 
have  25  :  100  :  :  10  :  40  feet,  the  velocity  of  the  water 
at  erf;  and  if  the  area  at  efbe  50  feet,  then  50  :  25  : :  40 
:  20  feet,  the  velocity  at  ef,  the  canal  being  kept  con- 
tinually full. 

The  quantity  of  water  that  flows  through  a  pipe,  or  in  a 
canal  or  river,  at  any  part,  is  in  proportion  to  the  area 
multiplied  by  the  velocity  at  that  part. 

The  calculation  of  the  motion  of  rivers  is  often  of  the 
hig/iest  utility  to  the  engineer.  This  is  sometimes  done  by 
Jhe  employment  of  very  intricate  formulas,  but  such 
methods,  if  easier  could  be  found,  would  evidently  be  in- 

195 


196  HYDROl/rNAMICS. 

consistent  with  the  nature  of  this  work.  The  method  which 
we  shall  give  is  simple,  and  will  be  found  to  answer  all  the 
purposes  of  the  practical  man. 

In  a  river,  the  greatest  velocity  is  at  the  surface  and  in 
jthe  middle  of  the  stream  ;  from  which  it  diminishes  toward 
the  bottom  and  sides,  where  it  is  least. 

The  velocity  at  the  middle  of  the  stream  may  be  ascer 
tained,  by  observing  how  many  inches  a  body  floating  with 
the  current  passes  over  in  a  second  of  time.  Gooseberries 
will  fit  this  purpose  exceedingly  well;  if  they  are  not  at 
hand,  a  cork  may  be  employed. 

Take  the  number  of  inches  that  the  floating  body  passes 
over  in  one  second,  and  extract  its  square  root ;  double  this 
square  root,  subtract  it  from  the  velocity  at  top,  and  add 
one,  the  result  will  be  the  velocity  of  the  stream  at  the 
bottom. 

And  these  velocities  being  ascertained,  the  mean  velocity, 
or  that  with  which  if  the  stream  moved  in  every  part, 
it  would  produce  the  same  discharge,  may  be  found  = 
the  velocity  at  top  —  v/velocity  at  top  +  '5. 

Exam. — If  the  velocity  at  the  top  and  in  the  middle  of 
the  stream,  be  36  inches  per  second,  then,  36  —  (2  X  \/36) 
-f  1  =36  —  12  +  1  =  25  =  the  least  velocity,  or  the 
velocity  at  bottom.  And  the  mean  velocity  will  be  = 
36  —  v/36  -f  -5  =  36  —  6  +  -5  =  30-5. 

When  the  water  in  a  river  receives  a  permanent  increase 
from  the  junction  of  some  other  river,  the  velocity  of  the 
water  is  increased.  This  increase  in  velocity  causes  an  in- 
crease of  the  action  of  the  water  on  the  sides  and  bottom, 
from  which  circumstance  the  width  of  the  river  will  always 
be  increased,  and  sometimes,  though  not  so  frequently,  the 
depth  also.  By  the  reason  of  this  increased  action  of  the 
water  on  the  bottom,  the  velocity  is  diminished  until  the 
tenacity  of  the  soil  or  the  hardness  of  the  rock  afford  a 
sufficient  resistance  to  the  force  of  the  water.  The  bed  of 
the  river  then  changes  only  by  very  slow  degrees,  but  the 
bed  of  no  river  is  stationary. 

It  is  of  the  greatest  use  to  know  the  amount  of  the  action 
of  any  stream  on  its  bed,  and  for  this  purpose  a  knowledge 
of  the  nature  of  the  bed  and  of  the  velocity  at  bottom,  are 
absolutely  necess-ary. 

Every  kind  of  soil  has  a  certain  velocity  which  will  insure 
the  stability  of  the  bed.  A  less  velocity  would  allow  the 


HYDRODYNAMICS.  197 

waters  to  deposit  more  of  the  matter  which  is  carried  with 
the  current,  and  a  greater  velocity  would  tear  up  the  chan- 
nel. From  extensive  experiments  it  has  been  found,  that 
a  velocity  of  3  inches  per  second  at  the  bottom,  will  just 
begin  to  work  upon  the  fine  clay  used  for  pottery,  and, 
however  firm  and  compact  it  may  be,  it  will  tear  it  up. 
A  velocity  of  6  inches  will  lift  fine  sand — 8  inches,  will  lift 
coarse  sand  (the  size  of  linseed) — 12  inches,  will  sweep 
along  gravel — 24,  will  roll  along  pebbles  an  inch  diameter 
— and  3  feet  at  bottom,  will  sweep  along  shivery  stones  the 
size  of  an  egg. 

When  water  issues  through  a  hole  in  the  bottom  or  side 
of  a  vessel,  its  velocity  is  the  same  as  that  acquired  by  a 
body  falling  through  free  space  from  a  height  equal  to  that 
of  the  surface  of  the  water  above  the  hole. 

The  most  correct  rule  for  ascertaining  the  velocity  of 
water  running  through  pipes  and  canals  is  this  : 

57  X  height  of  head  X  diam.  of  pipe\  1 

Vlength  of  pipe  X  57  X  diam.  of  pipe/ 
the  velocity  in  inches  with  which  the  water  will  issue  from 
the  orifice.    All  the  measures  are  understood  to  be  taken  in 
inches. 

Exam. — If  there  be  a  reservoir  of  water  whose  depth  is 
6  feet,  having  a  tube  1  foot  long  and  2£  inches  bore,  open 
so  as  to  let  the  water  escape  at  a  distance  of  18  inches  from 
the  bottom,  then  we  have,  6  X  12  =  72  =  whole  depth 
of  water  on  the  reservoir,  and  72  —  18  =  54,  the  height 
of  the  head  of  the  fluid  above  the  orifice,  wherefore  by  the 
rule, 


v'  (4-5)  X  231  =  2-121  X  23s  =  49-49  inches  per  second, 
the  velocity  of  the  water.  And,  by  multiplying  this  result 
by  the  area  of  the  orifice,  we  get  the  quantity  discharged  in 
one  second — hence,  if  the  pipe  be  circular,  we  have, 

2-5  2-5  x  3-1416        . 

=  1-25  =  radius,  and =  naif 

^  £ 

tircu inference  =  1-9635  =  area  of  orifice,  hence,  49-49  X 
1-9635  =  97-173  cubic  inches. 

The* quantity  of  water  that  flows  out  of  a  vertical  rectan- 
gular aperture,  that  reaches  as  high  as  the  surface,  is  |  of 
17* 


198  HYDRODYNAMICS. 

the  quantity  that  would  flow  out  of  the   same  aperture, 
placed  horizontally  at  the  depth  of  the  tiase. 

When  water  issues  out  of  a  circular  aperture  in  a  thin 
plate  placed  on  the  bottom  or  side  of  a  reservoir,  the  stream 
is  contracted  into  a  smaller  diameter,  to  a  certain  distance 
from  the  orifice.  The  vein  is  smaller  at  the  distance  of  half 
the  diameter  of  the  orifice  where  the  area  of  the  section  .of 
the  vein  is  •{-{?•  of  that  of  the  orifice,  and  at  the  above  point 
the  stream  has  the  velocity  given  by  theory,  so  that  to  ob- 
tain the  quantity  of  water  discharged,  we  multiply  the 
velocity  by  the  area  of  the  orifice,  and  {°  of  this  will  be 
the  true  result.  -When  the  water  issues  through  a  short 
tube,  the  vein  of  the  stream  will  be  less  contracted  than  in 
the  former  case,  in  the  proportion  of  16  to  13.  But  when 
the  water  issues  through  an  aperture  which  is  the  frustum 
of  a  cone,  whose  greater  base  is  the  aperture,  the  height  of 
the  conic  frustum  =  one  half  the  diameter  of  the  aperture 
and  the  area  of  the  small  end  to  that  of  the  large  end,  as 
10  :  16 ;  then,  in  this  case,  there  will  be  no  contraction  of 
the  vein  ;  and  from  this  it  may  be  inferred,  that,  when  a 
supply  of  water  is  required,  the  greatest  possible  from  a 
given  orifice,  this  form  should  be  employed. 

To  determine  the  quantity  of  water  discharged  by  a 
small  vertical  or  horizontal  orifice,  the  time  of  discharge, 
and  the  height  of  the  fluid  in  the  vessel,  when  any  two 
of  these  quantities  are  known. 

Let  A  represent  the  area  of  the  small  orifice,  W  the 
quantity  of  water  discharged  ;  T  the  time  of  discharge,  H 
the  height  of  fluid  in  the  vessel,  and  #  =  16-087  feet,  the 
space  described  by  gravity  in  a  second. 

Then  we  have, 

W  =  2  x  A  x  *  N/#  X  H 

4. 


2 

X 

t 

X 

W 

x  H 

ws 

H  = 


4  x  g  X  t2  X  A3 

By  means  of  these  formulas  we  may  determine  the  quan- 
tity of  water  W  which  is  discharged  in  the  same  time  T, 
from  any  other  vessel  in  which  A'  is  the  area  of  the  orifice, 


HYDRODYNAMICS. 


199 


and  H  the  altitude  of  the  fluid  ;  for  since  t  and  g  are  con 
etant,  we  shall  have 

W  :  W'=  A,/  H  :  A'  -,/  H'. 

Table  shou'iag  the  quantify  of  JTater  discharged  in  one 
Minute  by  Orifices  differing  in  form  and  position. 


aslant 

H(aiL-!it  of  the 
Fluiil  above 
the  centre  of 
the  orifice. 

Form  and  position  of  the  Orifice. 

Diameter  of 
the  orifice. 

No.  of  cubic 
inches  dis- 
charged in  a 
minute. 

Ft.    in.   lin. 

Lines. 

11  8  10 

Circular  and  Horizontal, 

6 

2311 

Circular  and  Horizontal, 

12 

9281 

Circular  and  Horizontal, 

24 

37203 

Rectangular   and    Hori- 

- 

zontal, 

12  by  3 

2933 

Horizontal  and  Square, 

12  side 

11817 

Horizontal  and  Square, 

24  side 

47361 

900 

Vertical  and  Circular, 

G 

2018 

Vertical,  and  Circular, 

12 

8135 

400 

Vertical  and  Circular, 

6 

1353 

Vertical  and  Circular, 

12 

5436  , 

507 

Vertical  and  Circular, 

12 

628 

From  these  results  we  may  conclude, 

1.  That  the  quantities  of  water  discharged  in  equal  times 
by  the  same  orifice  from  the  same  head  of  water,  are  very 
nearly  as  the  areas  of  the  orifices  ;  and, 

2.  That  the  quantities  of  water  discharged  in  equal  times 
by  the  same  orifices  under  different  heads  of  water,  are 
nearly  as  the  square  roots  of  the  corresponding  heights  of 
the  water  in  the  reservoir  above  the  centres  of  the  orifices. 

3.  The  quantities  of  water  discharged  during  the  same 
time  by  different  apertures  under  different  heights  of  water 
in  the  reservoir,  are  to  one  another  in  the  compound  ratio 
of  the  areas  of  the  apertures,  and  of  the  square  roots  of 

'the  heights  in  the  reservoirs. 

This  general  rule  may  be  considered  as  sufficiently  cor- 
rect for  ordinary  purposes  ;  but,  in  order  to  obtain  a  great 
Jflegree  of  accuracy,  Bossut  recommends  an  attention  to 
»he  three  following  rules. 

I.    Friction  is  the  cause,  that,  of  several  similar  orifices 


200 


HYDRODYNAMICS. 


the  smaLest  discharges  less  water  in  proportion  than  those 
which  are  greater,  under  the  same  altitudes  of  water  in  the 
reservoir. 

2.  Of  several  orifices  of  equal  surface,  that  which  has  the 
smallest  perimeter  ought,  on  account  of  the  friction,  to  give 
more  water  than  the  rest,  under  the  same  altitude  of  water 
in  the  reservoir. 

3.  That,  in  consequence  of  a  slight  augmentation  which 
the  contraction  of  the  fluid  vein  undergoes,  in  proportion 
as  the  height  of  fluid  in  the  reservoir  increases,  the  expense 
ought  to  be  a  little  diminished. 

Table  of  Comparison  of  the  Theoretic  with  the  Real  dis 
charges  from  an  orifice  one  inch  in  diameter. 


Constant 
height  of  the 
water  in  the 
reservoir 
above  the 
centre  of  the 
orifice. 

Theoretical  dis- 
charges through 
acircular  orifice 
one  inch  in  dia- 
meter. 

Real  discharges 
in  the  same  time 
through  the 
same  orifice. 

Ratio  of  the  theoretical 
to  the  real  discharges. 

Paris  feet. 

Cubic  inches. 

Cubic  inches. 

1 

4381 

2722 

1  to  0-62133 

2 

6196 

3846 

1  to  0-62073 

3 

7589 

4710 

1  to  0-62064 

4 

8763 

5436 

1  to  0-62034 

5 

9797 

6075 

1  to  0-62010 

6 

10732 

6654 

1  to  0-62000 

7 

11592 

7183 

1  to  0-61965- 

8 

12392 

7672 

1  to  0-61911 

9 

13144 

8135 

1  to  0-61892 

10 

13855 

8574 

1  to  0-61883 

11 

14530 

8990 

1  to  0-61873 

12 

15180 

9384 

1  to  0-61819 

13 

15797 

9764 

1  to  0-61810 

14 

16393 

10130 

1  to  0-61795 

15 

16968 

10472 

1  to  0-61716 

1 

2 

3 

4 

It  appears  from  this  table,  that  the  real  as  well  as  the 
theoretical  discharges  are  nearly  proportional  to  the  square 
roots  of  the  heights  of  the  fluid  in  the  reservoir.  Thus 
for  the  heights  1  and  4,  whose  square  roots  are  as  1  to  2 
feet,  the  real  discharges  are  2722  and  5436,  which  are  to 
one  another  as  1  to  1-997,  very  nearly  as  1  to  2. 


HYDRODYNAMICS.  SOi 

Let  it  be  required  to  determine  the  quantity  of  water  dis- 
charged from  an  orifice  of  3  inches  in  diameter,  under  an 
altitude  of  30  feet.  Then,  since  the  real  quantities  dis- 
charged are  in  the  compound  ratio  of  the  orifices,  and  the 
square  roots  of  the  altitudes  of  the  water,  and  since  the 
theoretical  discharge  by  an  orifice  1  inch  in  diameter,  under 
an  altitude  of  15  feet  is  16968  cubical  inches  in  a  minute, 
we  have  1  ^/  15 :  9  v/  30  =  16968:  215961,  the  theoreti- 
cal discharge.  But  the  theoretical  is  to  the  real  discharge 
as  1  to  -62,  the  above  value  being  diminished  in  that  ratio, 
gives  133309  cubic  inches  for  the  real  quantity,  of  water 
discharged  by  the  orifice. 

The  following  formulae  have  been  given  by  M.  Prony 
as  deduced  from  the  preceding  experiments  of  Bossut, 

Q  =  0-61938  AT  ^/  2  g  H, 

A  being  the  area  of  the  orifice  in  square  feet,  H  the  altitude 
of  the  fluid  in  feet,  T  the  time,  g  the  force  of  gravity  at 
the  end  of  a  second,  and  Q  the  quantity  of  water  in  cubic 
feet.  As  ^/  2  g  is  a  constant  quantity,  and  is  equal  to 
7*77125,  we  have 

Q  =  4-818  AT  */  H  for  orifices  of  any  form. 
If  the  orifices  are  circular,  and  if  d  represents  their  dia- 
meter, then 

Q  =  3-7842  d*  T  </  H. 
From  the  second  of  these  equations  we  obtain 

A  Q 

Till  T  </  H 

T. 5 

4-818  A  v/  H 

H  Q 

(4-818  AT)* 

These  formulae  will  be  found  to  give  very  accurate  r* 
suits  ;  but  if  we  wish  to  obtain  a  still  higher  degree  of  ac- 
curacy, we  must  not  use  the  mean  co-efficient  0*6194,  but 
the  one  in  the  table  which  comes  nearest  to  the  circum- 
stances of  the  case.  Thus  if  the  head  of  water  happens  tc 
be  small,  such  as  1  foot,  then  we  must  take  the  co-efficient 
fl-62133,  and  if  it  happens  to  be  great,  we  must  use  the  least 
'co-efficient  0-61716. 


202 


HYDRODYNAMICS. 


Table  containing  the  quantity  of  Water  discharged  over  a  weir. 


Depth  of  the  up- 
per  edije  of    the 
wasteboard  below 
the  surf-tee  in  Eng- 
lish  inches. 

Cubic   feet  of    water 
discharged  in  a  minute 
b_y    every  inch  of  the 
wasteboard.accorJii.g 
to  Du  Buat's  formula. 

Cubic  fe  t  of  water  dis- 

board  ace  >rding  to  expe- 
riments ir  adu  in  Scotland. 

1 

0,403 

0,428 

2 

1,WO 

1,211 

3 

2,095 

2,226 

4 

3,225 

3,427 

5 

4,507 

4,789 

6 

5,925 

6,295 

7 

7,466 

7,933 

8 

9,122 

9,692 

9 

10,884 

11,564 

10 

12,748 

13,535 

11 

14,707 

15,632 

12 

16,758 

17,805 

13 

18,895 

20,076 

14 

21,117 

22,437 

15 

23,419 

24,883 

16 

25,800 

27,413 

17 

28,258 

30,024 

18 

30,786 

32,710 

Talk  containing  the  quantities  of  Water  discharged  by  Cylindrical 
Tubes  one  inch  in  diameter  and  of  different  lengths,  whether  the 
Tubes  were  inserted  in  the  bottom  or  in  the  sides  of  the  vessel. 


Constant  altitude  of  the  fluid  above  the  superior  base  of  the  tube 
11  feet  8  inches  and  10  lines. 

Lengths  of  the  Tubes  expressed  in 
lines. 

charged  in  a  minute. 

The  tube  filled  with  the  \  jj® 
issuing  fluid  f 
J  iy 
The  tube  not  filled  Avith  £  - 
the  issuing  fluid  3 

12274 
12188 
12168 

9282 

IIYDKODVNAMICS. 


203 


Table,  of  comparison  of  lite  Theoretical  u-itft  the  Rial  Dixtliurgefi  from 
an  additional  Tube  of  a  cylindrical  form,  one  inch  in  diameter  ana 
tico  inches  long. 


foment    alli-u.lt 

TJieorrtical   dis- 

Ri-il  ilivhargei  in  the 

it  the    Wi-n    in 

chir^.5     'hrnn^li    a 

*  ilnc:    tinir   l.y    >  cy- 

Ratio  of  the  throrcticil 

itHivr  !(]<•    (t,  «n. 

circular  orifice  one 
nu  h  ID  diameter. 

in    iliiiin-'ir    .iii.l  livo 

to  the  real  discharge* 

of  the  i<n&ce. 

iuch«long. 

t  tris  ft-ll. 

Cubic-  inrhn. 

Cubic  iii.-lii-s. 

1 

4381 

3539 

1  to  0-81781 

2 

0190 

5002 

1  to  0-80729 

3 

7589 

6126 

1  to  0-80724 

4 

8763 

7070 

1  to  0-80681 

5 

9797 

7900 

1  to  0-80638 

6 

10732 

8654 

1  to  0-80638 

7 

11592 

9340 

1  to  0-80573 

& 

12392 

9975 

1  to  0-80496 

9 

13144 

10579 

1  to  0-80485 

10 

13855 

11151 

1  to  0-80483 

11 

14530 

11693 

1  to  0-80477 

12 

15180 

12205 

1  to  0-80403 

13 

15797 

12699 

1  to  0-80390 

14 

16393 

13177 

1  to  0-80382 

15 

16968 

13620 

1  to  0-80270 

1 

2 

3 

4 

Hence  it  follows,  that  the  velocity  in  English  inches  will 
be  V  =  22-47  v/  H  for  additional  tubes. 

M.  Prony  has  given  the  following  formulae,  as  deduced 
from  the  preceding  table. 


\  4-9438  'I 


4-9438  T  v/  H 

H-  Q 

~  (4-9438  d*  T)a 

The  resistance  that  a  body  sustains  in  moving  through  a 
fluid  is  in  proportion  to  the  square  of  the  velocity. 

The  resistance  that  any  plane  surface  encounters  in  rnov- 
irfg  through  a  fluid  with  any  velocity,  is  equal  to  the  weight 
of  a  column  whose  height  is  the  space  a  body  would  have 


'«J04  HYDRODYNAMICS. 

to  fall  through  in  free  space  to  acquire  that  velocity,  and 
whose  base  is  the  surface  of  the  plane. 

Ex. — If  a  plane  16  inches  square,  move  through  water  at 
the  rate  of  13  feet  per  second  ;  then, 
132 
64~  = 

the  space  a  body  would  require  to  fall  through  free  space  to 
acquire  a  velocity  of  13  per  second,  wherefore,  as  2'6  feet 
=31-2  inches,  we  have  16  x  31-2  =  499-2  cubic  inches= 
the  column  of  matter  whose  height  and  base  are  required ; 
therefore,  since  1728  cubic  inches  =  1  cubic  foot  of  water 
weighs  1000  ounces,  we  have  1728  :  499-2  :  :  1000  :  288 
ounces  =  18  Ibs.  which  is  the  amount  of  resistance  met 
with  by  the  plane  at  the  above  velocity. 

As  action  and  reaction  are  equal  and  contrary,  it  is  the 
same  thing  whether  the  plane  moves  against  the  fluid,  or 
the  fluid  against  the  plane. 


WATER  WHEELS. 

MOTION  is  generally  obtained  from  water,  either  by  ex- 
posing obstacles  to  the  action  of  its  current,  or  by  arresting 
its  progress  during  part  of  its  descent,  by  movable  buckets. 

Water-wheels  have  three  denominations  depending  on 
their  particular  construction,  undershot,  breast,  and  over- 
shot. If  the  water  is  to  act  on  the  wheel  by  its  weight,  it 
is  delivered  from  the  spout  as  high  on  the  wheel  as  possible, 
that  it  may  continue  the  longer  to  press  the  buckets  down ; 
but  when  it  acts  on  the  wheel  by  the  velocity  of  the  stream, 
it  is  made  to  act  on  the  float-boards  at  as  low  a  point  as 
possible,  that  it  may  have  acquired  previously  the  greatest 
velocity.  In  the  first  case,  the  wheel  is  said  to  be  overshot, 
in  the  second,  undershot.  The  overshot  wheel  is  the  most 
advantageous,  as  from  the  same  quantity  of  water  it  gives  a 
greater  power,  but  it  is  not  always  that  we  can  employ  an 
overshot  wheel  from  the  smallness  of  the  fall.  When  this 
is  the  case,  we  must  deliver  the  water  farther  down  than 
the  top  of  the  wheel,  and,  in  this  case,  it  becomes  a  breast- 
wheel,  and  partakes  in  some  degrees  of  the  properties  of 
the  overshot.  When  we  cannot  employ  a  breast  wheel, 
we  must  have  recourse  to  the  undershot,  which  is  the  least 


WATER    WHEELS. 


205 


powerful  of  all.  The  force  of  a  stream  of  water  against 
the  floats  of  an  undershot  wheel  is  equal  to  a  column  of 
water,  whose  base  is  the  section  of  the  stream  in  that  place, 
and  height  the  perpendicular  height  of  the  water  to  the 
surface.  Where  the  quantity  of  water  is  given,  its  force 
against  the  floats  of  the  wheel  is  directly  proportional  to 
the  velocity  of  the  stream,  or  the  square  root  of  the  heigh! 
of  the  surface.  These  remarks  hold  true  only  when  the 
water  is  allowed  to  escape  from  the  float  boards,  after  it 
has  struck  them.  For  if  the  floats  be  too  near  each  other, 
thcnihe  water  from  one  float  will  be  sent  back  and  obstruct 
the  progress  of  the  next  float. 

Engraved  representations  of  the  three  forms  of  the  water 
wheel  are  given  in  plate  1st.  Fig.  1  is  a  representation 
of  the  undershot ;  fig.  2  of  the  breast ;  and  fig.  3  of  the 
overshot  water  wheel.  The  floats  of  the  undershot  as 
likewise  of  the  breast  wheel  are  flat,  those  of  the  latter 
being  fitted  so  nearly  to  the  water  way  that  little  of  the 
fluid  is  allowed  to  escape  between  their  edges  and  the  stone 
or  brick  work,  as  may  be  seen  in  the  figure.  The  over- 
shot wheel  is  furnished  with  buckets  instead  of  floats,  so 
constructed  that  they  shall  retain  as  much  as  possible  of 
the  water  from  the  time  they  receive  it  until  they  arrive  al 
the  lowest  point,  where  each  bucket  should  be  emptied, 
since  if  any  water  be  carried  by  the  bucket  in  its  ascent  it 
will  be  just  so  much  unnecessary  weight  that  the  wheel  has 
to  lift.  The  following  geometrical  construction  will  show 
the  method  of  forming  the  buckets  so  that  there  shall  be  the 
greatest  possible  advantage  derived  from  the  overshot  wheel. 

This  bucket  is  formed  of  three  planes  ; 
AB  is  in  the  direction  of  the  radius  of 
the  wheel,  and  is  called  the  start,  or 
shoulder  ;  BC  is  called  the  arm,  and  CH 
the  u'ritf.  These  buckets  are  so  con- 
structed, that  when  AB  makes  an  angle 
of  35°  with  the  vertical  diameter  of  the 
.•wheel,  the  line  AD  is  horizontal ;  and  the 
area  of  the  figure  ADCB  is  equal  to  that 
tofFCBA;  so  that  as  much  water  is  re- 
tained in  the  bucket  in  this  position  as  would  fill  FCBA ; 
the  whole  of  the  water  is  not  discharged  until  CD  becomes 
horizontal,  which  takes  place  when  the  line  AB  is  very 
near  the  lowest  point. 

18 


206  HYDRODYNAMICS. 

To  find  the  velocity  of  the  water  acting  upon  the  wheel, 
^(height  of  the  fall  x  64-38)—  the  velocity  in  feet  per 
second. 

Ex. — If  the  height  of  the  fall  be  14  feet,  then  we  have 
*/(14x64-38)  =  .v/ 901-32=30-02  feet  per  second,  nearly. 

To  find  the  area  of  the  section  of  the  stream, 

The  number  of  feet  flowing  in  1  second 

velocity  in  feet  per  second 
the  section  of  the  stream  in  square  feet. 

Ex. — If  there  be  40  feet  flowing  in  a  second,  and  the 
velocity  of  the  stream  is  5  feet  per  second,  then, 

i°=8  = 
5 

the  area  of  the  section  of  the  stream  in  square  feet. 

To  calculate  the  power  of  the  fall : 

Area  of  section  of  stream  where  it  acts  upon  the  wheel  X 
height  of  fall  x  62|  =  the  number  of  Ibs.  avoir,  the  wheel 
can  sustain,  acting  perpendicularly  at  its  circumference,  so 
as  to  be  in  equilibrium.  If  this  number  of  Ibs.  which  keeps 
the  wheel  at  rest  be  diminished,  the  wheel  will  move. 

If  the  wheel  move  as  fast  as  the  stream,  it  is  clear  that 
the  water  would  have  no  effect  in  moving  it, — if  the  wheel 
were  to  move  faster  than  the  stream,  the  water  would  -be  a 
positive  hindrance  to  its  motion  ;  and  it  can  only  be  ad- 
vantageous when  the  velocity  of  the  stream  is  greater  than 
that  of  the  wheel.  There  is  a  certain  relation  between  the 
velocity  of  the  wheel  and  that  of  the  stream,  at  which  the 
effect  will  be  the  greatest  possible  or  a  maximum. 

The  effect  of  an  undershot  wheel  is  a  maximum  when 
the  velocity  of  the  wheel  is  5  of  the  velocity  of  the  stream. 

Ex. — If  the  area  of  the  cross  section  of  a  stream  be  6 
feet,  and  its  velocity  4  feet  per  second,  and  a  fall  of  16  feet 
can  be  procured,  then  we  have  4x6=24,  the  number  of 
cubic  feet  flowing  per  second  : 

v/  (16x64-38)=32,  the  velocity  of  the  water  at  the  end 
of  the  fall :  . 

— =  I,  the  section  of  the  stream  at  the  end  of  the  fall  in 

o2 

square  feet : 

|  X  16  X  62|  =  750  lbs.=  the  weight  which  the  wheel 
will  sustain  in  equilibrium 


WATER    WHEELS.  207 

Now,  the  effective  velocity  of  the  stream  is  the  difference 
between  the  velocities  of  the  stream  and  wheel,  and  the 
wheel's  velocity  being  3  of  that  of  the  stream,  the  difference 
or  effective  velocity  will  be  § ;  now,  the  power  of  the 
stream  is  as  the  square  of  the  effective  velocity,  and  the 
square  of  f  is  -J.  We  must  multiply  the  power  of  the  fall 
as  above  calculated  by  this  £,  and  also  by  5,  in  order  that 
the  wheel  may  move  with  the  proper  velocity  ;  hence,  750 
X  £  xl  =  lll'  Ibs.  raised  through  10|  feet  per  second,  the 
velocity  of  the  wheel,  which  is  5  of  32  the  velocity  of  the 
stream.  An  undershot  water  wheel  is  capable  only  of  rais- 
ing 2^-  of  the  weight  of  the  water  to  the  height  of  the  fall. 
From  numerous  experiments  on  water  wheels,  it  has  been 
found,  that  in  practice  the  water  not  being  allowed  to  es- 
cape from  the  floats  immediately  after  it  has  impinged  upon 
them,  the  maximum  effect  is,  when  the  velocity  varies  be- 
tween j  and  I,  that  of  the  water  being  nearly  ^.  There 
is  another  deviation  from  theoretical  result,  in  consequence 
of  the  water  not  being  allowed  to  escape  immediately  from 
the  float-boards,  as  the  water  is  heaped  up  to  about 2|  times 
its  natural  height,  and  thus  acts  partly  by  its  weight,  and 
partly  by  its  force — in  consequence  of  which  it  happens, 
that  a  well-constructed  undershot  water  wheel,  instead  of 
raising  *24T  of  the  weight  of  the  water  expended  on  the 
height  of  the  fall,  will  raise  3. 

The  effective  head  being  the  same,  the  effect  of  the  wheel 
will  depend  on  the  quantity  of  water  expended ;  and  the 
quantity  of  water  being  the  same,  the  'effect  of  the  wheel 
depends  on  the  height  bf  the  head  of  the  fall. 

The  section  of  the  stream  being  the  same,  the  effect  will 
be  nearly  as  the  cube  of  the  velocity. 

Overshot  water  wheel. — If  th*  water  in  the  buckets  of  an 
overshot  wheel  be  supposed  to  be  equally  diffused  over  half 
the  circumference  of  the  wheel,  then  the  whole  weight  of 
the  water  in  the  buckets  is  to  its  power  to  turn  the  wheel 
as  1 1  to  7. 

,  An  overshot  water  wheel  will  raise  nearly  as  much  water 
to  the  height  of  the  fall,  as  is  expended  in  driving  the 
•tfheel :  if  the  height  of  the  fall  be  reckoned  from  the  bucket 
that  receives  the  water  to  the  bucket  that  discharges  if. 
According  to  the  last  experiments,  the  velocity  of  an  over- 
shot wheel  should  be  between  2  and  4  feet  per  second  5or 


208  HYDRODYNAMICS. 

all  diameters  of  wheels.     A  breast  wheel  partakes  of  the 
properties  of  the  two  foregoing,  as  part  of  its  action  de 
pends  on  the  velocity,  and  part  on  the  weight  of  the  water 
which  moves  it. 

Circumstances  will  regulate  which  of  these  three  species 
of  water  wheels  is  to  be  employed.  For  a  large  supply  of 
water  with  a  small  fall,  the  undershot  wheel  is  the  most  ap- 
propriate. For  a  small  supply  of  water  with  a  large  fall,  the 
overshot  ought  to  be  employed.  Where  both  the  quantity 
of  water  and  height  of  fall  are  moderate,  the  hreast  wheel 
must  be  used. 

Before  erecting  a  water  wheel,  all  the  circumstances  must 
be  taken  into  account,  and  our  calculations  made  accordingly. 
We  must  measure  the  height  of  head  velocity,  and  area  of 
stream,  &c.,  to  do  which  a  slight  knowledge  of  levelling  will 
be  required.  What  follows  will  make  this  subject  suffi- 
ciently plain. 

Levelling. — A  pole  about  10  feet  long  must  be  procured, 
:md  also  a  staff  about  five  feet  long,  on  the  top  of  which  is 
fixed  a  spirit  level  with  small  sight  holes  at  the  ends,  so 
that  when  the  spirit  level  is  perfectly  horizontal,  the  eye 
may  view  any  object  before  it  through  the  sights  in  a  per- 
fectly horizontal  line.  If  you  have  to  measure  the  perpen- 
dicular distance  between  the  bottom  and  top  of  a  hill,  for 
instance  ;  place  the  level  staff  on  the  side  of  the  hill  in  such 
a  way  that  when  the  level  is  truly  set,  the  top  of  the  hill 
may  be  seen  through  the  sights.  Keep  the  level  in  this 
position  and  look  the  contrary  way,  then  cause  some  person 
to  place  the  10  feet  staff  before  the  sight  further  down  the 
hill,  and  looking  through  the  sights  to  the  staff,  cause  the 
person  to  move  his  finger  up  or  down  the  staff  until  the 
finger  be  seen  through  the  sights,  and  mark  the  position  of 
the  finger  on  the  staff.  Keep  your  ten  feet  staff  in  the 
same  place,  and  carry  your  level  staff  down  the  hill  to  a 
convenient  distance,  then  fix  it  in  the  same  way  as  before; 
and  looking  through  the  sights  at  the  ten  feet  staff,  cause 
the  person  to  bring  his  finger  towards  the  bottom  of  the 
staff,  and  move  his  finger  up  or  down  the  staff  in  the  same 
way  until  it  be  seen  through  the  sights,  and  mark  the  place 
of -the  finger.  Then  the  distance  between  the  two  linger 
marks  added  to  the  height  of  the  level  staff,  will  be  the 
perpendicular  distance  between  the  place  where  the  level 
staff  now  stands  and  the  top  of  the  hill.  The  process  is 


WATER    WHEELS. 


209 


perfectly  simple,  and  it  will  not  be  difficult  to  repeat  it 
oftener  if  the  height  of  the  hill  requires  it. 

This  process  will  give  what  is  called  the  apparent  level, 
which  however  is  not  the  true  level.  Two  stations  are  on 
the  same  true  level  when  they  are  equally  distant  from  the 
centre  of  the  earth.  The  apparent  level  gives  the  objects 
in  the  same  straight  line,  but  the  true  level  gives  the  line 
which  joins  them  as  part  of  a  circle  whose  centre  is  the 
centre  of  the  earth.  In  small  distances  there  is  no  sensible 
difference  between  the  true  and  apparent  level  of  any  two 
objects.  When  the  distance  is  one  mile,  the  true  level  will 
be  about  8  inches  different  from  the  apparent  level.  This 
will  serve  well  enough  to  remember,  but  more  correctly 
speaking  it  is  7*962  inches  for  one  mile,  and  for  all  other 
distances  the  difference  of  the  two  levels  will  be  as  the 
square  of  the  distance.  Thus  at  the  distance  of  two  miles 
u  will  be, 

I3  :  2s  : :  8  :  32  inches,  or  2  feet  8  inches  nearly. 

These  circumstances  must  be  strictly  observed  in  the 
formation  of  canals,  railways,  &c.,  &c. 

The  following  table  will  save  the  trouble  of  calculation. 
The  distances  are  measured  on  the  earth's  surface. 


Distance 
yards. 

Allowance 
in 
inches. 

Distance 
measured  in 
miles. 

Allowance  iu 
fert  and 
inches. 

100 

0-026 

k 

0     0 

200 

0-103 

5 

0     2 

300 

0-231 

1 

0     4 

400 

0-411 

1 

0     8 

500 

0-643 

2 

2     8 

600 

0-925 

3 

6     0 

700 

1-260 

4 

10     7 

800 

1-645 

5            16     7 

900 

2-081 

6 

23  11 

1000 

2-570 

7 

32     6 

1100 

3-110 

8 

42     6 

1200 

3-701 

9 

53     9 

1300 

4^344 

10 

66     4 

1400 

5-038 

11 

80     3 

1500 

5-784 

12 

95     7 

1600 

6-580 

13 

112     2 

1700 

7-425 

14 

130     1 

IS* 


21C  HYDRODYNAMICS. 

Construction  of  a  water  wheel.  —  To  find  the  centre  of 
gyration  of  a  water  wheel,  take  the  radius  of  the  whee] 
and  the  weight  of  its  arms,  rim,  shrouding,  and  float 
boards.  Then  call  the  weight  of  the  rim  R,  which  must 
be  multiplied  by  the  square  of  the  radius,  and  the  pro- 
duct be  doubled  and  then  carried  out.  Next  the  weight 
of  the  arms  called  A  must  be  multiplied  by  the  square 
of  the  radius,  and  be  doubled  and  carried  out  as  before. 
Then  the  weight  of  the  water  in  action  called  W  must 
be  multiplied  by  the  square  of  the  radius  and  carried 
out.  If  these  products  be  added  together  into  one  sum 
they  will  form  a  dividend.  For  a  divisor,  double  the  sum 
of  the  weights  of  the  rim  and  the  arms,  and  add  the  weight 
of  the  water  to  them.  Divide  the  dividend  by  the  divisor. 
and  the  square  root  of  the  quotient  will  be  the  radius  of 
gyration. 

Ex.  —  In  a  wheel  24  feet  diameter  —  The  weight  of  the 
arms  is  2  tons,  the  shrouding  and  rims  4  tons,  and  the 
water  in  action  2  tons  ;  hence,  by  the  above, 

R  =  4  tons  x  12a  X  2  =  1152 
A  =  2  tons  x  12a  x  2  =  576 
W  =  2  tons  x  123  =  288 

Their  sum         2016  dividend,  and 
2  X  (4  -f  2  +  2)  =  16,  the  divisor. 

The  answer,  J^^h  =  «/126  =  11-225. 


Tables  for  the  more  ready  performance  of  calculations 
for  water  wheels  are  usually  given  in  books  of  Mechanics  ; 
the  construction  and  use  of  which  we  shall  now  proceed  to 
explain. 

1.  Find,  by  measuring  and  levelling,  the  height  of  the 
fall  of  water  which  is  reckoned  from  its  upper  surface  to 
the  middle  of  the  depth  of  the  stream,  where  it  acts  upon 
the  float-boards. 

2.  Find  the  velocity  acquired  by  the   water  in  falling 
through  that  height,  which   is   done   thus  :    multiply  the 
height  of  the  fall  by  64-38,  extract  the  square  root  of  the 
product  which  would  be  the  velocity  of  the  stream  if  there 
were  no  friction,  but  to  allow  for  friction  take  away  •£•§  of 
this  result  for  tke  true  velocity. 


WATKK    WHEELS.  211 

3.  Find  the  velocity  that  ought  to  be  gi\en  to  the  float- 
Doards,  by  taking  4  °f  lne  velocity  of  the  water,  which 
product  will  be  the  number  of  feet  the  float-boards  have  to 
pass  through  in  one  second  of  time  to  produce  the  maxi- 
mum effect. 

circumference  of  wheel 
velocity  of  the  float-boards 

the  number  of  seconds  that  the  wheel  takes  to  make  one 
turn. 

4.  Divide  60  by  the  last  number.     The  quotient  is  the 
number  of  revolutions  the  wheel  makes  in  one  minute. 

5.  Divide  90  by  the  last  quotient,  the  new  quotient  is  the 
number  of  turns  of  the  millstone  for  one  of  the  wheel :    90 
being  the  number  of  turns  that  a  millstone  of  five  feet  dia- 
meter ought  to  make  in  a  minute. 

6.  As  the  number  of  turns  of  the  wheel  in  a  minute 

Is  to  the  number  of  turns  of  the  millstone  in  a  minute, 
So  is  the  number  of  staves  in  the  trundle 
To  the  number  of  teeth  in  the  spur-wheel,  avoiding 
fractions. 

7.  The  number  of  turns  of  the  wheel  in  a  minute    x 
the   number   of  turns    of  the   millstone    for   one  turn  of 
the   wheel    =   the  number  of   turns  of  the  millstone  per 
minute. 

Or,  by  a  different  method,  multiply  the  number  of  teeth 
in  the  spur-wheel  by  the  number  of  turns  of  the  water- 
wheel  per  minute,  and  divide  this  product  by  the  number 
of  staves  in  the  trundle,  the  quotient  is  the  number  of  turns 
of  the  millstone  per  minute. 

In  this  way  has  the  following  table  been  constructed  for 
a  water-wheel  of  1 5  feet  diameter,  the  millstone  being  5 
feet  diameter  and  making  90  tu»ns  in  one  minute. 


213 


HYDRODYNAMICS 


A    MILLWRIGHT  S     TABLE, 


In  which  the  Velocity  of  the  Wheel  is  Three-Seve.it/is  of  the  Velocity 
of  the  Wafer,  allowance  being  made  fur  Ihe  Effects  <-f  Friction  en 
the  Velocity  of  the  Stream  for  a  Wheel  of  Fifteen  Feet  diameter. 


r 

Height  of 
the  fall  of 
water. 

Velocity  of 
the  watei 
per  second. 

Velocity  of 
wheel  per 
second, 
hein*  It-Ttlii 
of  that  of 
the  water. 

Revolutions 
nf  Ihe 

"nmutT 

Numlier  of 

revolutions  i  f 

for  one  of 

UM 

Uhr.l. 

Ti-eth  in  Ihe 
wheel,  and 

»':«•!•»  in  Ihe 
trundle. 

R.-V)  utions 
of  the  mill- 

minute  bv 
(he*catavgi 

,-ind  teeth. 

« 

•3 

£-£  = 

A  *    . 
S'~5 

J=  Ji 

1^1 

1 

£2.£ 
8 

IfJ 

u.  g.- 

8 

•f  3  "3 

Ifi 

res 

c  g.- 

£§£ 

*| 

3  c  = 

>  s.? 

Oj  O   U 

PS  2  u 

1 

7-62 

3-27 

4-16 

21-63 

130     6 

90-07 

2 

10-77 

4-62 

5-88 

15-31 

92     6 

90-16 

3 

13-20 

5-66 

7-20 

12-50 

100     8 

90-00 

4 

15-24 

6-53 

8-32 

10-81 

97     9 

89-67 

5 

17-04 

7-30 

9-28 

9-70 

97   10 

90-02 

6 

18-67 

8-00 

10-19 

8-83 

97   11 

89-86 

7 

20-15 

8-64 

10-99 

.  8-19 

90  11 

89-92 

8 

21-56 

9-24 

11-76 

7-65 

84  11 

89-80 

9 

22-86 

9-80 

12-47 

7-22 

72   10 

89-68 

10 

24-10 

10-33 

13-15 

6-84 

82   12 

89-86 

11 

25-27 

10-83 

13-79 

6-53 

85  13 

90-16 

12 

26-40 

11-31 

14-40 

6-25 

75  12 

90-00 

13 

27-47 

11-77 

14-99 

6-00 

72   12 

89-94 

14 

28-51 

12-22 

15-56 

5-78 

75   13 

89-77 

15 

29-52 

12-65 

16-13 

5-58 

67   12 

90-06 

16 

30-48 

13-06 

16-63 

5-41 

65   12 

90-06 

17 

31-42 

13-46 

17-14 

5-25 

63   12 

89-99 

18 

32-33 

13-86 

17-65 

5-10 

61   12 

89-72 

19 

33-22 

14-24 

18-13 

4-96 

60   12 

90-65 

20 

34-17 

14-64 

18-64 

4-83 

58   12 

90-09 

It  is  desirable  that  the  millwright  should  possess  short 
easy  rules,  which  would  answer  the  purposes  of  practice 
rather  than  the  conditions  of  mere  theory.  The  following 
will  be  found  useful,  as  they  give  the  power  with  allowance 
for  friction  and  waste  of  water. 


WATER    WHEELS.  213 

For  an  undershot  : 

Height  of  fall  X  quantity  of  water  flowing  per  minute 

DOOQ 
the  number  of  horses'  power  which  the  effect  is  equal  to. 

For  an  overshot  : 

Power  of  an  undershot  X  2|  =  horses'  power. 

For  a  breast-wheel  : 

Find  the  power  of  an  undershot  from  the  top  of  the  fall 
to  where  the  water  enters  the  bucket  ;  then  for  an  overshot 
for  the  rest  of  the  fall  —  the  sum  of  these  two  is  the  power 
of  the  breast  wheel. 

NOTE.  —  The  quantity  of  water  flowing  per  minute,  ana 
the  height  of  the  fall  are  both  taken  in  feet. 

Ex.  —  What  power  can  be  obtained  from  an  undershot 
wheel  —  the  fall  being  25  feet,  the  section  of  the  stream 
being  9  feet,  and  the  velocity  of  the  water  18  feet  per 
minute? 

9  x  18   X  25       4050 


5000  5000 

one  horse  power  being  unit. 

And  an  overshot  in  the  same  situation  would  be  '81  X 
2-5  ==  2-025  horses'  power. 

And  if,  in  a  breast  wheel,  the  water  enters  the  bucket  10 
.reet  from  the  top  of  the  fall,  then  we  have, 

10  X  8   X  9      ol  _     720       o,  _  1800-0  _ 
~~5000~  ~~5000X  :    5000    = 

for  an  overshot,  and  for  the  undershot  we  found  it  before 
81  ;  hence,  '36  +  '81  =  1-17  horses'  power  for  the  breast 
wheel. 

BARKER'S  MILL. 

IN  plate  1st,  fig.  4,  we  have  given  a  view  of  Barker's 
mill,  where  CD  is  a  vertical  axis,  moving  on  a  pivot  at  D, 
and  carrying  the  upper  millstone  m,  after  passing  through 
an  opening  in  the  fixed  millstone  C.  Upon  this  axis  is 
fixed  a  vertical  tube  TT  communicating  with  a  horizontal 
rube  AB,  at  the  extremities  of  which  A,  B,  are  two  aper- 
tures in  opposite  directions.  When  water  from  the  mill- 
course  MN  is  introduced  into  the  tube  TT,  it  flows  out  of 
the  apertures  A,  B,  and  by  the  reaction  or  counterpressure 
of  the  issuing  water,  the  arm  AB,  and  consequently  the 


214  HYDRODYNAMICS. 

whole  machine,  is  put  in  motion.  The  bridge-tree  ab  is 
elevated  or  depressed  by  turning  the  nut  c  at  the  end  of 
the  lever  cb.  In  order  to  understand  how  this  motion  is 
produced,  let  us  suppose  both  the  apertures  shut,  and  the 
tube  TT  filled  with  water  up  to  T.  The  apertures  A,  B, 
which  are  shut  up,  will  be  pressed  outwards  by  a  force 
equal  to  the  weight  of  a  column  of  water  whose  height  is 
TT,  and  whose  area  is  the  area  of  the  apertures.  Every 
part  of  the  tube  AB  sustains  a  similar  pressure ;  but  as 
these  pressures  are  balanced  by  equal  and  opposite  pres- 
sures, the  arm  AB  is  at  rest.  By  opening  the  aperture  at 
A,  however,  the  pressure  at  that  place  is  removed,  and 
consequently  the  arm  is  carried  round  by  a  pressure  equal 
to  that  of  a  column  TT,  acting  upon  an  area  equal  to  that 
of  the  aperture  A.  The  same  thing  happens  on  the  arm 
TB  ;  and  these  two  pressures  drive  the  arm  AB  round  in 
the  same  direction.  This  machine  may  evidently  be  ap- 
plied to  drive  any  kind  of  machinery,  by  fixing  a  wheel 
upon  the  vertical  axis  CD. 

This  ingenious  machine  has  not  been  much  employed, 
even  in  those  situations  for  which  it  is  best  adapted  ;  partly, 
we  suspect,  from  the  millwright's  riot  having  in  his  posses- 
sion sufficiently  simple  rules  for  its  construction ;  as  the 
theory  of  Barker's  mill,  simple  as  its  construction  and  action 
may  appear,  is  not  by  any  means  well  developed.  In  the 
mean  time  the  following  directions  may  be  found  useful  to 
the  mechanic. 

1.  Make  each  arm  of  the  horizontal  tube,  from  the  centre 
of  motion  to  the  centre  of  the  aperture  of  any  convenient 
length,  not  less  than  ^  of  the  perpendicular  height  of  the 
water's  surface  above  these  centres. 

2.  Multiply  the  length  of  the  arm  in  feet  by  -61365,  and 
the  square  root  of  this  product  will  be  the  proper  time  for  a 
revolution,  in  seconds ;   then  adapt  the  other  parts  of  the 
machinery  to  this  velocity ;  or, 

If  the  time  of  a  revolution  be  given,  multiply  the  square 
of  this  time  by  1-6296  for  the  proportional  length  of  the 
arm  in  feet. 

Multiply  together  the  breadth,  depth,  and  velocity  per 
second  of  the  race,  and  divide  the  last  product,  14-27  X 
the  square  root  of  the  height ;  the  result  is  the  area  of 
either  aperture  ; — or,  multiply  the  continual  product  of  the 
breadth,  depth,  and  velocity  of  the  race,  by  the  square  root 


BARKER'S  MILL.  215 

of  the  height,  and  by  the  decimal  -07,  the  last  product 
divided  by  the  height  will  give  the  area  of  the  aperture. 

Multiply  the  area  of  either  aperture  by  the  height  of  the 
head  of  water,  and  this  product  by  55*795  (or,  in  round 
numbers,  56)  for  the  moving1  force,  estimated  at  the  centres 
of  the  apertures  in  Ibs.  avoirdupois. 

Ex. — If  the  fall  be  18  feet  from  the  head  to  the  centre 
of  the  apertures,  then  the  arm  must  not  be  less  than  2  feet, 
as^of  18  =  2,  and,/ (2  x  -61305)  =  v/(  1-22730)  =  1-107 
—  the  time  of  a  revolution  in  seconds;  also,  the  breadth 
of  the  race  being  17  inches,  and  depth  9,  and  the  velocity 
of  the  water  6  feet  per  second,  here  we  have, 

17  in.  =  1-41  feet,  and  9  in.  =  -75  feet,  then 

1-41  x  '75  x  6  =  6-34  =  the  area  of  section  of  the 
race  x  velocity  of  water ;  hence, 

6-39  x  N/  18  X  -07  =  1-896  =  the  area  of  the  aperture 
in  inches ;  and, 

1-876  x  18  X  56  =  1909  Ibs.  the  moving  force. 

The  following  dimensions  have  been  employed  in  prac- 
tice with  success.  The  length  of  arm  from  the  centre  pivot 
to  the  centre  of  the  discharging  hole,  46  inches  ;  inside 
diameter  of  the  arm,  13  inches  ;  diameter  of  the  supplying 
pipe,  2  inches  ;  height  of  the  working  head  of  water  21 
feet  above  the  level  of  the  discharge.  When  a  machine 
of  these  dimensions,  and  in  such  circumstances,  was  not 
loaded  and  had  one  orifice  open,  it  made  115  turns  in  a 
minute. 


PNEUMATICS. 


PNEUMATICS  comprehends  the  knowledge  of  the  proper- 
ties of  common  air  and  elastic  fluids  in  general. 

Air  is  capable  of  being  compressed  to  almost  any  degree, 
that  is,  may  be  forced  into  a  space  infinitely  smaller  than 
the  space  which  it  commonly  occupies,  and  this  is  effected 
by  additional  pressure.  When  this  additional  pressure  is 
taken  away,  the  air  will  regain,  by  its  elasticity,  its  former 
magnitude.  Were  it  not  for  this  circumstance,  the  subject 
of  this  chapter  might  have  been  introduced  when  we  dis- 
cussed the  equilibrium  and  motion  of  water  and  fluids, 
which  are  non-elastic  or  incompressible,  as  their  fundamen- 
tal laws  are  the  same.  It  has,  indeed,  been  found  oy  recent 
experimenters,  that  water,  mercury,  &c.,  are  compressible, 
but  to  a  very  limited  degree  ;  so  that  although  the  distinction 
of  elastic  and  non-elastic  fluids  is  not  absolutely  correct,  it 
is  yet  sufficiently  so  to  retain  Pneumatics,  in  elementary 
arrangement,  as  a  distinct  branch  of  science. 

The  air  or  atmosphere  is  a  fluid  body  which  surrounds 
(he  earth,  and  gravitates  on  all  parts  of  its  surface. 

The  mechanical  properties  of  air  are  the  same  as  other 
elastic  fluids,  and  being  the  most  common,  inquiries  in 
pneumatics  are  generally  confined  to  this  fluid. 

The  air  has  weight.  A  cubic  foot  of  it  weighs  1P2857 
ounces  at  the  surface  of  the  earth,  or,  as  some  state  it,  1'222. 

The  air  being  an  elastic  fluid,  it  is  compressible  and  ex- 
pansible, and  its  degrees  of  compression  and  expansion  are 
proportional  to  the  forces  or  weights  which  compress  it. 

All  the  air  near  the  earth's  surface  is  in  a  state  of  com- 
pression, in  consequence  of  the  weight  of  the  atmosphere 
which  is  above  it. 

As  the  less  weight  that  presses  the  air  compresses  it  the 
less,  or  causes  it  to  be  less  dense,  and  as  the  higher  we 
rise  in  the  atmosphere  there  will  be  the  less  weight,  so  the 
higher  we  go  in  the  atmosphere  the  air  will  be  the  less 
dense. 

2Jfi 


OF    THE    ATMOSPHERE.  217 

The  spring  or  elasticity  of  the  air  is  equal  to  the  weight 
of  the  atmosphere  above  it,  and  they  will  produce  the  same 
effects  since  they  always  sustain  and  balance  each  other. 

It'  the  density  of  the  air  be  increased  by  compression,  its 
spring  or  elasticity  is  also  increased,  and  in  the  same  pro- 
portion. 

By  the  pressure  and  gravity  of  the  atmosphere  on  the 
surface  of  fluids  such  as  water,  they  are  made  to  rise  in 
pipes  or  vessels,  where  the  spring  or  pressure  Within  is 
taken  off  or  diminished.  This  fact,  a  knowledge  of  which 
is  applied  to  a  multitude  of  useful  purposes,  will  uot  be 
difficult  of  explanation.  L*it  a  tube  3  feet  long  bo  filled 
with  water,  the  tube  being  open  at  one  end  and  close  at  the 
other;  one  unacquainted  witn  the  subject  might  naturally 
expect  that  if  this  tube  were  held  perpendicularly  with  the 
open  end  downmost,  the  water  would  flow  out  of  *he  tube 
by  reason  of  its  weight.  But  if  we  consider  all  the  c.ireum- 
stances,  we  will  see  that  this  can  only  happen  on  certain 
conditions.  The  water  has  a  tendency  to  fall  to  thr  earth 
in  consequence  of  its  weight,  but  then  the  air  of  the  aL:no- 
sphere,  which  we  have  stated  before  as  also  possessed  of 
weight,  presses  upon  the  surface  of  the  water  at  the  r.pen 
end  of  the  tube ;  and  as  the  pressure  of  fluids  of  t-Il  kinds 
is  exerted  in  every  direction,  it  follows,  that  the  \\r  will 
have  a  tendency  to  force  the  water  up  the  tube.  Now  the 
pressure  of  the  atmosphere  at  the  surface  of  the  earth  is 
about  15  Ibs.  for  every  square  inch,  which  is  therefore  the 
force  by  which  the  water  will  be  pressed  up  the  tube  by 
the  action  of  the  air.  A  column  of  water  3  feet  high  does 
not  exert  such  a  pressure  on  the  base ;  wherefore ,  as  the 
pressure  upwards  is  greater  than  the  pressure  downwards, 
the  water  will  remain  suspended  in  the  tube. 

Let  us  now  take  a  tube  36  feet  long,  similar  to  the 
former,  filled  with  water  and  inverted  in  the  same  way  as 
before,  it  will  now  be  found  that  a  part  of  the  water  will 
tlow  out  of  the  tube,  the  reason  of  which  will  be  easily  seen. 
,  It  was  stated  under  Hydrostatics,  that  the  pressure  of  a 
column  of  water  30  feet  high  was  equal  to  13  Ibs.  on  the 
/quare  inch.  So  that  we  see,  that  the  pressure  of  the  air 
will  keep  30  feet  of  the  water  in  the  tube,  but  it  will  keep 
more,  for  the  pressure  of  the  air  is  15,  and  that  of  30  feet 
of  water  is  only  13  ;  and  as  the  pressure  of  the  water  will 
be  as  its  depth,  we  say,  13  :  15  : :  30  :  34,  which,  there- 

19 


PNEUMATICS. 

fore,  is  the  greatest  height  at  which  the  water  will  be  sup- 
ported by  the  pressure  of  the  atmosphere. 

For  the  purpose  of  arriving  at  this  conclusion  of  the 
effect  of  the  pressure  of  the  atmosphere,  we  might  have 
employed  a  much  shorter  tube  if  we  had  used  a  heavier 
fluid  than  water,  for  instance,  mercury.  Now  the  cubic 
foot  of  mercury  weighs  13600  ounces,  and  a  cubic  inch  will 

be  found,  13600 

=  7-866  ounces, 


or  nearly  8  ounces,  that  is  about  half  a  pound  avoirdupois  ; 
therefore  30  inches  will  weigh  15  Ibs.,  hence,  the  atmo- 
sphere will  balance  by  its  pressure  30  inches  of  mercury. 
Thus  we  have  arrived  at  the  principle  of  the  barometer,  or 
weather  glass,  as  it  is  commonly  called.  The  pressure  of 
the  air  at  the  surface  of  the  earth  is  not  always  constant, 
but  varies  within  certain  limits.  The  mean  pressure  is 
about  14  Ibs.  to  the  square  inch,  and  the  corresponding 
height  of  the  mercury  in  the  barometer  will  therefore  be 
15  :  14  :  :  30  :  28  inches. 

It  will  appear  evident,  from  what  has  been  said  before, 
that  as  the  higher  we.  ascend  in  the  atmosphere  there  will 
be  less  pressure,  and  therefore  the  mercury  in  the  barometer 
will  fall,  and  this  fact  has  been  used  as  a  means  of  measur- 
ing heights  by  the  barometer.  If  the  air  were  of  the  same 
uniform  density  up  to  the  top  of  the  atmosphere  as  it  is  at 
the  earth's  surface,  we  might  very  easily  determine  its 
height,  for  the  specific  gravity  of  air  being  to  that  of  water 
as  1-222  to  1000,  nearly,  we  have  this  proportion,  1'222  : 
1000  :  :  33-25,  (the  mean  height  of  a  water  barometer  in 
feet,)  :  27200  feet,  which  is  very  nearly  5|  miles  ;  but  by 
a  process  which  proceeds  on  correct  principles,  .the  height 
of  the  atmosphere  has  been  estimated  at  about  50  miles. 
The  law  of  the  diminution  of  density  at  different  heights  in 
the  atmosphere  is  this,  that  if  the  heights  increase  in  arith- 
metical progression,  the  densities  will  decrease  in  geometri- 
cal progression;  for  instance,  if  the  density  at  the  surface 
of  the  earth  be  called  1,  and  if  at  the  height  of  7  miss  it 
be  called  4  times  rarer  than  at 

14  16, 

21   it  will  be  64  times  rarer, 

28  256, 

35  1024, 


PRESSURE    OK    THE    ATMOSPHERE.  219 

and  in  this  way  it  might  be  shown,  that  at  the  height  of  one- 
half  the  diameter  of  the  earth,  one  cubic  inch  of  atmospheric 
air  of  the  density  which  we  breathe,  would  expand  so  much 
as  to  fill  the  bounds  of  the  solar  system. 

Many  eminent  men  have  investigated  this  subject,  and 
derived  theorems  of  great  use  for  determining  altitudes  by 
the  barometer.  Some  of  these  are  exceedingly  complex 
and  unfitted  for  a  work  of  this  nature  :  that  of  Sir  J.  Leslie 
is  the  most  simple,  and  gives  results  sufficiently  near  the 
truth  for  all  ordinary  purposes. 

As  the  sum  of  the  heights  of  the  mercury  at  the  bottom 
and  top  of  the  mountain  is  to  the  difference  of  the  heights, 
so  is  52000  to  the  altitude  of  the  mountain  in  feet. 

At  the  bottom  of  a  hill  the  barometer  stood  at  29'8,  and 
at  the  top  27*2,  wherefore, 

29-8  +  27-2  =  57  =  the  sum, 
and  29-8  —  27-2  =  2-6  =  the  difference  ; 
hence,  57  :  2*6  :  :  52000  :  2372  feet,  the  height  of  the 
•mountain  nearly. 

When  air  becomes  denser,  its  elastic  force  is  increased, 
and  that  in  proportion.  Thus,  when  air  is  compressed  into 
half  its  bulk,  its  elastic  force  will  be  double  of  what  it  was 
before. 

It  will,  therefore,  be  easy  to  calculate  the  elastic  force 
of  air  compressed  any  number  of  times ; — thus,  if,  by  any 
means,  we  condense  the  air  in  a  vessel  into  £  of  the  space 
which  it  occupied  when  not  confined,  it  will  press  on  the 
inside  of  the  vessel  with  a  force  of  15  X  3  =  45  Ibs.  on 
every  square  inch.  It  must  be  remembered,  however,  that 
the  atmosphere  presses  with  a  force  of  15  Ibs.  on  each  square 
inch  of  the  outside  of  the  vessel,  which  therefore  counter- 
acts so  much  of  the  force  of  the  condensed  air  within — the 
real  pressure,  therefore,  is  45  —  15  =  30  Ibs.  It  is  clear, 
then,  that  whatever  be  the  degree  of  condensation  of  the 
enclosed  air,  we  must  always  deduct  the  pressure  of  the  at- 
mosphere to  ascertain  its  true  effect.  The  young  mechanic 
will  easily  understand  what  is  meant  by  the  phrase — a 
pressure  of  2,  3,  4,  or  any  number  of  atmospheres,  one 
Atmosphere  being  understood  as  exerting  a  pressure  of  15 
Ibs.  on  the  square  inch,  two  atmospheres  30,  and  three  45, 
&c.  When  the  air  is  by  any  means  entirely  taken  out  of 
sny  vessel,  there  is  said  to  be  a  vacuum  in  that  vessel. 
What  is  the  whole  amount  of  pressure  on  the  inside  sur- 


220  PNEUMATICS. 

(ace  of  a  sphere,  which  contains  air  condensed  to  |  of  its 
natural  bulk,  and  is  6  inches  in  diameter  within.  Here, 
by  mensuration,  we  have,  6-  X  3-1416  =  113-0976  =  the 
surface  of  the  inside  of  the  sphere — and  15  x  4  —  15  = 
45  =  the  pressure  on  a  square  inch,  therefore,  113-0976  X 
45  =  5089-3920  Ibs.  on  the  inner  surface  of  the  globe. 
Here  the  globe  is  supposed  to  be  in  a  vacuum. 

In  a  cylinder  6  feet  long,  and  closed  at  the  bottom,  a  pis- 
ton is  thrust  down  to  the  distance  of  one  foot  from  the  bot- 
tom, the  cylinder  being  24  inches  in  diameter,  then,  by  the 
rules  in  mensuration,  the  area  of  the  piston  will  be  found  to 
be  452-4  inches,  the  diameter  of  the  piston  being  24  inches, 
and  the  cylinder  being  6  feet  long,  and  the  piston  being 
pressed  down  to  1  foot  from  the  bottom,  the  air  will  be  com- 
pressed into  -g-  of  its  former  bulk,  and  its  elastic  force  will 
be  6  times  greater  than  it  was  before.  At  first  it  was  15  Ibs. 
to  the  square  inch,  but  now  it  will  be  15  X  6  =  90  on  the 
square  inch,  and  one  atmosphere  being  deducted  for  the 
contrary  pressure  of  the  atmosphere  above  the  piston,  the 
pressure  is  90  —  15  =  75  Ibs.  to  the  square  inch,  where- 
fore, 452-4  x  75  =  33930  Ibs.,  the  force  by  which  the 
piston  will  be  pressed  upwards. 

THE    SYPHON. 

A  SYPHON,  or,  as  it  is  frequently  written,  siphon,  is  any 
bent  tube. 

If  a  syphon  be  filled  with  water  and  inverted,  so  that  the 
bend  shall  be  uppermost,  then  if  the  legs  be  of  equal 
length,  and  it  be  held  so  that  the  two  lower  ends  of  the 
syphon  are  on  a  level,  then  we  will  find  that  if  the  perpen- 
dicular height  of  the  bend  of  the  tube  above  the  level  of  the 
two  ends  be  not  more  than  32  or  33  feet,  the  water  will  re- 
main suspended  in  the  tube.  It  will  not  be  difficult  to  see 
how  this  happens,  for  the  atmosphere  pressing  on  the  water 
at  the  orifice  of  the  tube  at  each  extremity,  presses  the 
water  up  the  tube  with  a  force  capable  of  raising  it  33  feet ; 
but  in  the  case  supposed,  the  orifices  and  the  legs  are  equal, 
and  do  not  exceed  the  limit  of  32  or  33  feet,  therefore, 
since  the  pressure  on  one  orifice  is  the  same  as  the  pressure 
on  the  other,  there  will  be  an  equilibrium — and  the  water 
in  the  one  leg  has  no  more  power  to  move  than  that  in  the 
other. 

If  we  now  suppose  the  syphon  to  be  inclined  a  little,  so 


PUMPS.  221 

that  the  twc  orifices  shall  not  be  on  a  level,  or  what  is  the 
same  thing,  if  we  suppose  the  length  of  the  one  leg  to  be 
greater  than  that  of  the  other,  we  will  find  that  the  equi- 
librium will  be  no  longer  maintained  ;  and  the  water  will 
flow  out  of  the  orifice  which  is  lowest.  For  although  the 
air  presses  equally  on  both  orifices  with  a  force  of  15  Ibs. 
to  the  square  inch,  yet  the  contrary  pressures  downwards 
by  the  weight  of  the  water  are  not  equal,  therefore  motion 
will  ensue  where  the  power  of  the  water  is  greatest.  If 
the  shorter  leg  be  immersed  in  a  vessel  of  water,  and  the 
syphon  be  set  a  running,  the  water  will  flow  out  of  the 
lower  end  of  the  syphon,  until  the  other  end  be  no  longer 
supplied.  Instead  of  filling  the  syphon  with  water,  as  has 
been  supposed  above,  a  common  practice  is  to  apply  the 
mouth  to  the  lower  orifice,  and  by  sucking,  exhaust  the  air 
in  the  tube,  which  diminishes  the  pressure  at  the  other 
orifice,  and  consequently  the  action  of  the  atmosphere  will 
force  the  water  in  the  vessel  up  the  tube  of  the  syphon  and 
fill  it,  and  it  will  continue  to  act  in  the  same  way  as  before. 

PUMPS. 

A  PUMP  is  a  machine  used  for  exhausting  vessels  con- 
taining air,  or  for  raising  water,  sometimes  by  means  of 
the  pressure  of  the  atmosphere,  sometimes  by  the  condensa- 
tion of  air,  and  sometimes  by  a  combination  of  both. 

It  may  be  necessary  here  to  explain  what  is  meant  by  the 
term  valve,  that  our  remarks  on  the  action  of  the  pump  may 
be  rendered  more  intelligible. 

A  valve  is  usually  defined  to  be  a  close  lid  affixed  to  a 
tube  or  opening  in  a  vessel,  by  means  of  a  hinge  or  some 
sort  of  movable  joint,  and  which  can  be  opened  only  in  one 
direction.  There  are  various  kinds  of  valves.  The  clack 
valve  consists  merely  of  a  circular  piece  of  leather  covering 
tlje  hole  or  bore  of  the  pipe  which  it  is  intended  to  stop, 
.and  moving  on  a  hinge,  sometimes  a  part  of  itself,  and 
sometimes  made  of  metal.  The  butterfly  valve,  which  is 
superior  to  the  clack  valve,  consists  of  two  pieces  of  leather 
each  formed  into  the  shape  of  a  half  circle ;  they  are  at- 
Aached  by  hinges  on  their  diameters,  or  straight  parts,  to  a 
bar  that  crosses  the  centre  of  the  orifice  to  be  closed.  The 
button  or  conical  valve  consists  of  a  plate  of  brass  ground 
in  such  a  way  as  exactly  to  fit  the  conical  cavity  in  which  it 
lies.  Sometimes  valves  are  made  in  the  form  of  pyramids 

19* 


222 


PNEUMATICS 


IB 


consisting  of  four  triangular  flaps  which  form  the  "sides  of 
the  pyramid,  and  move  upon  hinges  which  are  placed  round 
the  edge  of  the  orifice  to  be  closed.  The  tops  of  these 
flaps  must  all  meet  accurately  in  the  middle  of  the  orifice 
and  are  supported  by  four  bars  which  meet  in  the  centre. 

The  action  of  the  air  pump  may  be  thus  explained.  Le' 
R  be  the  section  of  a  glass 
bell,  called  a  receiver,  closed 
at  the  top  T,  but  open  at  the 
bottom,  and  having  its  lower 
edge  ground  smooth,  so  as 
to  rest  in  close  contact  with 
a  smooth  brass  plate,  of 
which  SS  is  a  section.  In 
the  middle  is  an  opening  A, 
which  communicates  by  a- 
tube  AB  with  a  hollow  cy- 
linder or  barrel,  in  which  a 
solid  piston  P  is  moved. 
The  piston  rod  C  moves  in 
an  air-tight  collar  D,  and  at 
the  bottom  of  the  cylinder  a  valve  V  is  placed,  opening 
freely  outward,  but  immediately  closed  by  any  pressure 
from  without.  There  is  thus  a  free  communication  between 
the  receiver  R,  the  tube  AB,  and  the  exhausting  barrel  BV. 
When  the  piston  CP  is  pressed  down,  and  has  passed  the 
opening  at  B,  the  air  in  the  barrel  BV  will  be  enclosed,  and 
will  be  compressed  by  the  piston.  As  it  will  thus  be  made 
to  occupy  a  smaller  space  than  before,  its  density,  and  con- 
sequently its  elasticity,  will  be  increased.  It  will  therefore 
press  downwards  upon  the  valve  V  with  a  greater  force 
than  that  by  which  the  valve  is  pressed  upwards  by  the 
external  air.  This  superior  elastic  force  will  open  the 
valve,  through  which,  as  the  piston  descends,  the  air  in  tke 
barrel  will  be  driven  into  the  atmosphere.  If  the  piston  be 
pushed  quite  to  the  bottom,  the  whole  air  in  the  barrel  will 
be  thus  expelled.  The  moment  the  piston  begins  to  ascend, 
the  pressure  of  the  air  from  without  closes  the  valve  V  com- 
pletely, and,  as  the  piston  ascends,  a  vacuum  is  left  beneath 
it;  but,  when  it  rises  beyond  the  opening  B,  the  air  in  the 
receiver  R  and  the  tube  AB  expands,  by  its  elasticity,  so 
as  to  fill  the  barrel  BV.  A  second  depression  of  the  piston 
will  expel  the  air  contained  in  the  barrel,  and  the  process 


PUMPS.  223 

may  be  continued  at  pleasure.  The  communication  be- 
tween the  barrels  and  the  receiver  may  be  closed  by  a  stop- 
cock at  G.  In  consequence  of  the  elasticity  of  the  air  it 
expands  and  fills  the  barrel,  diffusing  itself  equally  through- 
out the  cavity  in  which  it  is  contained.  The  degree  of 
rarefaction  produced  by  the  machine  may,  hojivever,  be 
easily  calculated.  Suppose  that  the  barrel  contains  one- 
third  as  much  as  the  receiver  and  tube  together,  and,  there- 
fore, that  it  contains  one-fourth  of  the  whole  air  within  the 
valve  V.  Upon  one  depression  of  the  piston,  this  fourth 
part  will  be  expelled,  and  three-fourths  of  the  original  quan- 
tity will  remain.  One-fourth  of  this  remaining  quantity 
will  in  like  manner  be  expelled  by  the  second  depression 
of  the  piston,  which  is  equal  to  three-sixteenths  of  the  ori- 
ginal quantity.  By  calculating  in  this  way,  it  will  be  found 
that  after  thirty  depressionsAf  the  piston,  only  one  3096th 
part  of  the  original  quantity  will  be  left  in  the  receiver. 
Rarefaction  may  thus  be  carried  so  far  that  the  elasticity  of 
the  air  pressed  down  by  the  piston  shall  not  be  sufficient  to 
force  open  the  valve. 

We  now  proceed  to  the  consideration  of  the  common 
suction  pump.  This  pump  consists  of  a  hollow  cylinder  A, 
of  wood 'or  metal,  which  contains  a  piston  B, 
stuffed  so  as  to  move  up  or  down  in  the  cy- 
linder easily,  and  yet  be  air  tight :  to  this 
piston  there  is  attached  a  rod  which  will  reach 
at  least  to  the  top  of  the  cylinder  when  the 
piston  is  at  the  bottom.  In  the  piston  there 
is  a  valve  which  opens  upwards,  and  at  the 
bottom  of  the  cylinder  there  is  another  valve 
C  also  rising  upwards,  and  which  covers  the 
orifice  of  a  tube  fixed  to  the  bottom  of  the  cy- 
linder, and  reaching  to  the  well  from  whence 
the  water  is  to  be  drawn.  This  tube  is  commonly  called 
the  suction  tube,  and  the  cylinder,  the  body  of  the  pump. 

From  what  has  been  said  of  the  pressure  of  the  atmo 
sphere,  it  will  not  be  difficult  to  understand  how  this  ma- 
diine  operates.  For  when  the  piston  is  at  the  bottom  of  the 
cylinder,  there  can  be  no  air,  or  at  least  very  little,  between 
it  and  the  valve  C,  for  as  the  piston  was  pushed  clown,  the 
valve  in  it  would  allow  the  air  to  escape  instead  of  being 
condensed,  and  when  it  is  drawn  up,  the  pressme  of  the  air 
would  shut  'he  valve,  and  there  would  be  a  vacuum  produced 


ffl 


224  PNEUMATICS. 

in  the  body  of  the  cylinder  when  the  piston  arrived  at  the 
top.  But  the  air  in  the  cylinder  being  very  much  rarifisd, 
the  pressure  of  the  valve  C  on  the  water  at  the  bottom  will 
be  greatly  less  than  that  of  the  external  atmosphere  on  the 
surface  of  the  water  in  the  well  ;  therefore,  the  water  will 
be  pressed  up  the  pump  to  a  height  not  exceeding  32  or 
33  feet.  As  the  valves  shut  downwards,  the  water  is  pre- 
vented from  returning,  and  the  same  operation  being 
repeated,  the  water  may  be  raised  to  any  height,  not 
exceeding  the  above  limit,  in  any  quantity. 

The  quantity  of  water  discharged  in  a  given  time,  is  de- 
termined by  considering  that  at  each  stroke  of  the  piston  a 
quantity  is  discharged  equal  to  a  cylinder  whose  base  is  the 
area  of  a  cross  section  of  the  body  of  the  pump,  and  height 
the  play  of  the  piston.  Thus,  if  the  diameter  of  the  cylin- 
der of  the  pump  be  4  inches^nd  the  play  of  the  piston  3 
feet,  then,  by  mensuration,  we  have  to  find  the  content  of  a 
cylinder  4  inches  diameter,  and  3  feet  high — now,  4  inches 
is  the  f  of  a  foot,  or  -333,  hence,  -333a  x  '7854  =  -110999 
X  '7854  =  -08796  =  the  area  of  the  cross  section  of  the 
cylinder  in  square  feet;  hence,  -08796  X  3  =  -2639  =  the 
content  of  the  cylinder  in  cubic  feet  =  the  quantity  in 
cubic  feet  of  water  discharged  by  one  stroke  of  the  piston. 
Now,  a  cubic  foot  of  water  weighs  about  63-5  Ibs.  avoirdu- 
pois, wherefore,  -2639  x  63-5  =  16-756  Ibs.  avoirdupois, 
and  an  imperial  gallon  is  equal  to  10  Ibs.  of  water  ;  whence, 
dividing  the  above  number  16-756  by  10,  we  get  the  num- 
ber of  ale  gallons  =  1-6756.  The  piston,  throughout  its 
ascent,  has  to  overcome  a  resistance  equal  to  the  weight  of 
a  column  of  water,  having  the  same  base  as  the  area  of  the 
piston,  and  a  height  equal  to  the  height  of  the  water  in  the 
body  of  the  pump  above  the  water  in  the  well. 

In  our  calculations  of  the  effects  of  the  pump,  it  will  be 
necessary  to  determine  the  contents  of  pipes,  for  which 
purpose  the  following  simple  rules  will  serve. 

Diameter  of  pipe  in  inches8  =  number  of  avoirdupois 
pounds  contained  in  3  feet  length  of  the  pipe. 

If  the  last  figure  of  this  be  pointed  off  as  a  decimal,  the 
result  will  be  the  number  of  ale  gallons,  and  if  there  be 
only  one  figure  this  is  to  be  considered  as  so  many  tenths 
of  an  ale  gallon  :  ale  gallons  x  282  =  the. number  of  cubic 
inches. 

Thus,  in  a  pipe  5  inches  diameter,  we  have, 


PUMPS.  225 

59  =  25  =  number  of  avoirdupois  pounds  contained  in  3 
feet  of  the  pipe  2-5  =  the  number  of  ale  gallons  and  2'5  x 
282  =  705  cubic  inches. 

These  rules  find  the  content  for  three  feet  in  length  of 
the  pipe,  the  content  for  any  other  length  may  be  found  by 
proportion;  thus,  for  a  pipe  6  inches  in  diameter,  and  IV 
feet  long  ;  we  have,  6a  =  30  =  pounds  of  water  avoir,  con- 
tained in  the  pipe  to  the  length  of  3  feet ;  therefore, 

3  :  12  :  :  36  :  144  =  the  number  of  pounds  in  12  feel 
length,  and, 

14-4  =  ale  gallons,  and  14'4  X  282  =  4060-8  =  the 
cubic  inches  in  12  feet  length. 

The  resistance  which  is  opposed  to  a  pump  rod  in  raising 
water,  is  equal  to  the  weight  of  a  column  of  water  whose 
base  is  the  area  of  the  piston,  and  height  the  height  of  the 
surface  of  the  water  in  the  body  of  the  pump  above  the 
surface  of  the  water  in  the  well,  together  with  the  friction 
and  the  piston  and  piston  rod. 

Suppose  the  body  of  the  pump  to  be  6  inches  in  diifmeter, 
and  the  height  to  which  the  water  is  raised  be  30  feet,  and 
also  the  weight  of  the  piston  and  rod  is  10  Ibs.,  and  the 
friction  is  -£•  of  the  whole  weight  of  the  water. 

Now,  fi8  =  36  =  the  Ibs.  avoirdupois  of  3  feet  of  the 
column  of  water,  but  the  column  is  30  feet,  therefore,  3  : 
30  :  :  36  :  360  Ibs.,  the  weight  of  the  whole  column.  To 
this  we  must  add  the  effect  of  friction,  which  we  have  sup- 
posed to  be  y  of  the  weight  of  the  water ;  hence, 

^fid 

' —  =  72  Ibs.,  and  this  must  be  added  to  the  weight  of 

D 

the  column  of  water,  which  gives  360  +  72  =  432  Ibs.  the 
whole  amount  of  resistance  arising  from  the  weight  of  the 
water  and  friction  ;  to  this  must  be  added  the  weight  of  the 
piston  and  pump  rod,  therefore,  432  -\-  10  =  442  =  the 
whole  resistance  opposed  to  the  rising  of  the  piston,  any 
thing  greater  than  this  will  raise  it. 

In  the  construction  of  pumps  it  is  usual  to  employ  a  lever 

•    to  work  the  piston,  which   gives  an  advantage  in  power  ; 

*and  if  in  the  case  estimated  above,  we  employ  a  lever  who? e 

'arms   are  in  the  proportion  of  10  to  1,  the  pump  might  te 

wrought  with  a  force  of  44*2  Ibs.,  or  we  may  say  45  Ibs. 

For  the  convenience  of  workmen  we  insert  the  following 
table.  It  has  been  calculated  on  the  supposition  that  the 
handle  of  the  pump  is  a  lever  which  srives  an  advantage  on 


22C 


PNEUMATICS. 


the  piston  rod  of  5  to  1,  and  that  a  man  can,  with  a  pump 
30  feet  long,  and  a  4  inch  bore,  discharge  27'5  wine  gallons 
(oil  measure)  in  a  minute.  And  if  it  be  required  to  find 
the  diameter  of  a  pump  that  a  man  could  work  with  the 
same  ease  as  the  above  pump  at  any  required  height  above 
the  surface  of  the  well,  this  table  will  give  the  diameter  of 
l-ore,  and  the  quantity  of  water  discharged  in  a  minute. 


Height  of  (lie  pump 
above  the  surfc.ee  of 
the  mill  in  feet. 

Diameter  of  the  bore 

where  the  piston 
works  iu  iuchei. 

Water  discharged  per 
gallons  and  pints. 

10 

6-93 

81     6 

15 

5-66 

54     4 

20 

4-90 

40     7 

25 

4-38 

32     6 

30 

4- 

27     2 

35 

3-70 

23     3 

40 

3-46 

20     3 

45 

3-27 

18     1 

50 

3-10 

16     3 

55 

2-95 

14     7 

60 

2-84 

13     5 

65 

2-72 

12     4 

70 

2-62 

11     5 

75 

2-53 

10     7 

80 

2-45 

10     2 

85 

2-38 

9     5 

90 

2-31 

9     1 

95 

2-25 

8     5 

100 

2-19 

8     1 

We  stated  before  that  water  could  not  be  raised  to  a 
greater  height  than  32  feet  by  means  of  the  kind  of  pump 
we  have  described,  and  it  may  seem  strange  that  this  table 
extends  to  100 ;  but  these  are  pumps  acting  on  a  different 
principle,  by  means  of  which  water  may  be  raised  to  any 
height,  and  whose  action  will  be  considered  before  we 
leave  this  subject. 

The  lifting  pump.  This  pump,  like  the  suction  pump,  has 

.  two  valves  and  a  piston,  both   opening  upwards  ;  but  the 

valve  in  the  cylinder,  instead  of  being  placed  at  the  bottom 

of  the  cylinder,  is  placed  in  the  body  of  it,  and  at  the  height 


PtJMPS. 


227 


where  the  water  is  intended  to  be  delivered.  The  bottom 
of  the  pump  is  thrust  into  the  well  a  considerable  way,,  and 
if  the  piston  be  supposed  to  be  at  the  bottom,  it  is  plain, 
that  as  its  valve  opens  upwards,  there  will  be  no  obstruction 
to  the  water  rising  in  the  cylinder  to  the  height  which  it  is 
in  the  well  ;  for,  by  the  principles  of  Hydrostatics,  water 
will  always  endeavour  to  come  to  a  level.  Now  when  the 
piston  is  drawn  up,  the  valve  in  it  will  shut,  and  the  water 
in  the  cylinder  will  be  lifted  up  ;  the  valve  in  the  barrel 
will  be  opened,  and  the  water  will  pass  through  it,  and  can- 
not return,  as  the  valve  opens  upwards  ;  —  another  stroke 
of  the  piston  repeats,  the  same  process,  and  in  this  way  the 
water  is  raised  from  the  well  :  but  the  height  to  which  it 
may  be  raised,  is  not  in  this,  as  in  the  suction  pump,  limited 
to  32  or  33  feet.  To  ascertain  the  force  necessary  to  work 
this  pump,  we  are  to  consider  that  the  piston  lifts  a  column 
of  water  whose  base  is  the  area  of  the.  piston,  and  height 
the  distance  between  the  level  of  the  water  in  the  well  and 
the  spout,  at  which  the  water  is  delivered.  Thus,  -if  the 
diameter  of  the  pump's  bore  be  4  inches,  and  the  height 
of  the  spout  above  the  level  of  the  well  =  40  feet,  then  we 
have  42  =  16  Ibs.  in  three  feet  of  the  barrel;  wherefore, 

3:40::16:213j  Ibs.  the  weight  of  the  water,  and  the 
friction  and  weight  of  the  piston  and  rod  must  be  added  to 
this,  to  find  the  whole  force  necessary.  If  the  friction  be 
reckoned,  as  it  usually  is,  •][•,  then  we  have, 


wherefore,  213  -|-  42  =  255;  as  we  have  neglected  frac- 
tions we  may  reckon  it  256,  and  if  the  weight  of  the  piston 
and  rod  be  20  Ibs.  the  whole  will  be  256  -f  20  =  276  Ibs.. 
the  whole  force  necessary  to  balance  the  piston  ;  any  thing 
greater  than  this  will  raise  it. 

The  forcing  pump  remains  to  be  consi- 
dered. The  piston  of  this  pump  has  no 
valve,  but  there  is  a  valve  at  the  bottom  of 
,the  cylinder,  the  same  as  seen  at  A.  In  the 
side  of  the  cylinder,  and  immediately  above 
tl/e  valve  B,  there  is  another  valve  A  open- 
ing outwards  into  a  tube  which  is  bent  up- 
wards to  the  height  H  at  which  the  water 
is  to  be  delivered.  When  the  piston  is 
raised,  the  valve  in  the  bottom  of  the  pump 


228  PNEUMATICS 

opens,  ana  a  vacuum  being  produced,  the  water  is  pressed 
up  into  the  pump  on  the  principle  of  the  sucking  pump 
But  when  the  piston  is  pressed  down,  the  valve  A  at  the 
bottom  shuts,  and  the  valve  B  at  the  side  which  leads  into 
the  ejection  pipe  opens,  and  the  water  is  forced  up  the  tube. 
When  the  piston  is  raised  again  the  valve  B  shuts,  and  the 
valve  A  opens.  The  same  process  is  repeated,  and  the 
water  is  thrown  out  at  every  descent  of  the  piston,  the  dis 
charge  therefore  is  not  constant. 

It  is  frequently  required  that  the  dis- 
charge from  the  pump  should  be  continu- 
ous, and  this  is  effected  by  fixing  to  the 
top  of  the  eduction  pipe  an  air  vessel. 
This  air  vessel  consists  of  a  box  AB,  in 
the  bottom  of  which  there  is  a  valve  C 
opening  upwards  into  the  box.  This  valve 
covers  the  top  of  the  eduction  pipe  D.  A 
tube,  E,  is  fastened  into  the  top  of  the  box, 
which  reaches  nearly  to  the  bottom  of  the  box,  it  rises  out 
of  the  box,  and  is  furnished  with  a  stop  cock.  If  the  stop 
cock  be  shut,  and  the  water  be  sent  by  the  action  of  the 
pump  into  the  air  vessel,  it  cannot  return  because  of  the 
shutting  of  the  valve  at  the  bottom  of  the  box  ;  and  be- 
cause of  the  space  occupied  by  the  water,  the  air  in  the  box 
is  condensed,  and  will  consequently  exert  a  pressure  on  the 
water  in  the  air  vessel.  If  the  water  fill  three-fourths  of 
the  box,  then  the  air  will  be  compressed  so  as  to  exert  four 
times  its  original  force ;  and  the  stop  cock  being  opened, 
the  water  will  be  forced  up  the  tube,  with  a  force  which 
will  send  it  one  less  than  as  many  times  32  feet  as  the  air 
is  compressed,  that  is,  in  the  case  supposed  3  X  32  =  96 
feet.  On  this  principle  it  is  that  jets  of  fountains  act. 

The  air  vessel  may  therefore  be  considered  as  a  magazine 
of  power,  and  so  long  as  there  is  as  much  water  forced  into 
the  fir  vessel  by  pumping,  as  there  is  forced  out  by  the 
pressure  of  the  air,  there  will  be  a  constant  jet  of  water. 

The  force  necessary  to  raise  the  piston  in  this  pump,  is 
found  exactly  in  the  same  way  as  for  the  suction  purnp. 
And  the  force  necessary  to  depress  the  piston,  is  found  by 
taking  the  weight  of  a  column  of  water,  whose  height  is 
equal  to  the  height  of  the  spout  where  the  water  is  de- 
livered above  the  level  of  the  piston,  before  it  begins  to 
descend  Thus,  if  the  piston  when  raised  is  26  feet  above 


WINDMILLS.  229 

the  level  of  the  well,  and  the  spout  is  63  feet  above  the  same 
level,  therefore,  the  height  of  the  column* is  63  —  26  =  37 
feet;  and  supposing  the  diameter  of  the  ejection  pipe  to 
be  5  inches,  we  have  for  3  feet  of  the  pipe  5a  =  25  Ibs.. 
•wherefore  for  37  feet  we  have, 

3  :  37  ::  25  :  308 1  Ibs. 

The  weight  of  the  piston  and  its  rods  oppose  the  raising  of 
the  piston,  but  assist  in  depressing  it. 

The  power  applied  to  the  piston  rod  of  a  suction  pump 
should  be  an  intermitting  power,  otherwise  there  will  be  a 
waste  of  .power ;  but  in  a  forcing  pump  the  power  must  be 
continued  throughout  equable.  A  single  stroke  steam  en- 
gine will  be  best  to  raise  water  by  the  sucking,  and  a 
double  stroke  by  a  forcing  pump.  The  piston  rod  of  a 
forcing  pump  should  be  loaded  with  a  weight  sufficient  to 
balance  a  column  of  water,  whose  base  is  the  section  of  the 
piston,  and  whose  height  is  the  excess  of  the  height  of  the 
spout  from  the  level  of  the  water  in  the  cistern  above  68 
feet.  This  will  cause  a  regular  application  of  power  when 
this  pump  is  wrought  with  a  steam  engine. 

WIND    AND    WINDMILLS. 

WE  have  seen  the  effect  of  the  pressure  of  air,  arising 
from  its  weight  and  elasticity  when  at  rest ;  it  now  remains 
for  us  to  consider  its  effects  when  put  in  motion,  as  in  the 
case  of  wind. 

Were  it  not  for  the  irregularity  in  direction  and  force  of 
the  wind,  it  would  be  the  most  convenient  of  all  the  first 
movers  of  machinery,  but  even  as  it  is,  its  efficacy  may  be 
taken  advantage  of,  and  deserves  our  consideration. 

The  force  with  which  wind  strikes  against  a  surface,  is 
as  the  square  of  the  velocity  of  the  wind.  This  simple 
theorem  is  so  nearly  true  that  it  may  be  employed  without 
fear  of  error. 

The  force  in  avoirdupois  pounds  with  which  the  wind 
strikes  on  any  surface  on  which  it  acts  perpendicularly  may 
be  found  by  using  the  rule, 

surface  struck  x  velocity  of  wind3  x  '002288  ; 
•\vhere  the  surface  and  velocity  of  wind  are  taken  in  feet, 
amd  the  time  1  second.     If  the  wind  moves  at  the  rate  of 
30  feet  per  second,  and  the  surface  exposed  to  it.«  action  be 
14  feet  square,  then,  14  x  303  x  '002288  =  28-8288. 

From  this  statement  it  might  appear  at  first  sight,  that  iu 
20 


830 


PNEUMATICS. 


the  case  of  mills  winch  act  by  the  impulse  of  wind  on  re-  . 
volving  surfaces  called  sails — it  might  appear,  we  say,  that 
the  greater  quantity  of  sail  exposed  to  the  action  of  the 
wind,  the  greater  would  be  the  effect  of  the  machine.  But 
this  has  been  found  not  to  hold  :  it  would  appear  that  the 
wind  requires  space  to  escape.  The  sails  of  the  windmill 
may  be  supposed  to  intercept  a  cylinder  of  wind ;  and  it 
would  seem,  that  when  the  whole  cylinder  is  intercepted, 
the  effect  of  the  machine  is  diminished  ;  and  it  is  concluded 
from  experiments,  that  the  sails  should  not  intercept  above 
seven-eighths  of  the  cylinder  of  the  wind. 

We  here  subjoin  a  tabular  view  of  the  effects  of  wind  at 
different  velocities. 

Table  showing  the  pressure  of  the  Wind  for  the  following  Velocities. 


Velocity  of  the  Wind. 

Force  upon  1  square  foot  in 
pounds  avoir. 

Miles  in  1  hour. 

Feet  in  1  second. 

1 

1-47 

•005 

2 

2-93 

•020 

3 

4-40 

•044 

4 

5-87 

•079 

5 

7-33 

•123 

10 

14-67 

•492 

15 

22-00 

1-107 

20 

29-34 

1-968 

25 

36-67 

3-075 

30 

44-01 

4-429 

35 

51-34 

6-027 

40 

58-68 

7-873 

45 

66-01 

9-963 

50 

73-35 

12-300 

60 

88-02 

17-715 

80 

117-36 

31-490 

100 

146-70 

49-200 

Windmills  are  constructed  either  so  that  the  sails  shall 
move  in  a  horizontal  plane,  or  in  a  plane  nearly  vertical ; 
their  former  are  called  horizontal,  and  the  latter  vertical 
windmills.  In  plate  2,,  fig.  1  and  2,  we  have  given  a  plan 
and  section  of  a  horizontal  windmill,  on  an  improved  con- 


WINDMILLS.  231 

struction.  HH  are  the  side  walls  of  an  octagonal  building 
which  contains  the  machinery.  These  walls  are  surmounted 
by  a  strong  timber  framing  GG,  of  the  same  form  as  the 
building,  and  connected  at  top  by  cross-framing  to  support 
the  roof,  and  also  the  upper  pivot  of  the  main  vertical 
shaft  AA,  which  has  three  sets  of  arms,  BB,  CC,  DD, 
framed  upon  it  at  that  part  which  rises  above  the  height 
of  the  walls.  The  arms  are  strengthened  and  supported 
by  diagonal  braces,  and  their  extremities  are  bolted  to 
octagonal  wood  frames,  round  which  the  vanes  or  floats  EE 
are  fixed,  as  seen  in  outline  in  fig.  2,  so  as  to  form  a  large 
wheel,  resembling  a  water  wheel,  which  is  less  than  the 
size  of  the  house  by  about  18  inches  all  round.  This  space 
is  occupied  by  a  number  of  vertical  boards  or  blinds  FF, 
turning  on  pivots  at  top  and  bottom,  and  placed  obliquely, 
so  as  to  overlap  each  other,  and  completely  shut  out  the 
wind,  and  stop  the  mill,  by  forming  a  close  case  surround- 
ing the  wheel ;  but  they  can  be  moved  altogether  upon 
their  pivots  to  allow  the  wind  to  blow  in  the  direction  of  a 
tangent  upon  the  vanes  on  one  side  of  the  wheel,  at  the 
time  the  other  side  is  completely  shaded  or  defended  by 
the  boarding.  The  position  of  the  blinds  is  clearly  shown 
at  FF,  fig.  2.  At  the  lower  end  of  the  vertical  shaft  AA, 
a  large  spur-wheel  aa  is  fixed,  which  gives  motion  to  a 
pinion  c,  upon  a  small  vertical  axis  rf,  whose  upper  pivot 
turns  in  a  bearing  bolted  to  a  girder  of  the  floor  n.  Above 
the  pinion  c,  a  spur-wheel  e  is  placed,  to  give  motion  to 
two  small  pinions  f,  on  the  upper  ends  of  the  spindles  g, 
of  the  millstone  n.  Another  pinion  is  situated  at  the 
opposite  side  of  the  great  spur-wheel  aa,  to  give  motion  to 
a  third  pair  of  millstones,  which  are  used  when  the  wind 
is  very  strong ;  and  then  the  wheel  turns  so  quick  as  not 
to  need  the  extra  wheel  e  to  give  the  requisite  velocity  to 
the  stones.  The  weight  of  the  main  vertical  shaft  is  borne 
by  a  strong  timber  6,  having  a  brass  box  placed  on  it  to 
receive  the  lower  pivot  of  the  shaft.  It  is  supported  at  its 
'ends  by  cross-beams  mortised  into  the  upright  posts  66,  as 
sjnown  in  the  plan,  fig.  2.  A  floor  or  roof  1 1  is  thrown 
across  the  top  of  the  brick  building  to  protect  the  machinery 
from  the  weather,  and  to  prevent  the  rain  blowing  down 
the  opening  through  which  the  shaft  descends,  a  broad  cir- 
cular hoop  K  is  fixed  to  the  floor,  and  is  surrounded  by 
another  hoop  or  case  L,  which  is  fixed  to  the  arms  DD  of 


23£  PNEUMATICS. 

the  wheel.  This  last  is  of  such  a  size,  as  exactly  to  go 
over  the  hoop  K,  without  touching  it  when  the  wheel  turns 
round.  By  this  means, .the  rain  is  completely  excluded 
from  the  upper  room  M,  which  serves  as  a  granary,  being 
fitted  up  with  the  bins  mm,  to  contain  the  different  sorts 
of  grain  which  is  raised  up  by  the  sack-tackle.  A  wheel  i 
is  fixed  on  the  main  shaft,  having  cogs  projecting  from  both 
sides.  Those  at  the  under  side  work  into  a  pinion  on  tho 
end  of  the  roller  K,  which  is  for  the  purpose  of  drawing 
up  sacks.  Another  pinion  is  situated  above  the  wheel  i, 
which  has  a  roller  projecting  out  over  the  flap-doors  seen 
at  p,  in  fig.  2,  to  land  the  sacks  upon.  The  two  pinions 
mm,  fig.  2,  are  turned  by  the  great  wheel  act,  and  are  for 
giving  motion  to  the  dressing  and  bolting  machines,  which 
are  placed  upon  the  floor  N,  but  are  not  shown  in  the 
drawing,  being  exactly  similar  to  the  dressing  machines 
used  in  all  flour-mills.  The  cogs  upon  the  great  wheel  a 
are  not  so  broad  as  the  rim  itself,  leaving  a  plain  rim  about 
three  inches  broad.  This  is  encompassed  by  a  broad  iron 
hoop,  which  is  made  fast  at  one  end  to  the  upright  post  b  ; 
the  other  being  jointed  to  a  strong  lever  n,  to  the  extreme 
end  of  which  a  purchase  o  is  attached,  and  the  fall  is  made 
fast  to  iron  pins  on  the  top  of  a  frame  fixed  to  the  ground. 
This  apparatus  answers  the  purpose  of  the  brake  or  gripe 
used  in  common  windmills  to  stop  their  motion.  By  pull- 
ing the  fall  of  the  purchase  0,  it  causes  the  iron  strap  to  em- 
brace the  great  wheel,  and  produces  a  resistance  sufficient 
to  stop  the  wheel.  The  mill  can  be  regulated  in  its  motion, 
or  stopped  entirely,  by  opening  or  shutting  the  blinds  F, 
which  surround  the  fan-wheel.  They  are  all  moved  at 
once  by  a  circular  ring  of  wood  situated  just  beneath  the 
lower  ends  of  the  blinds  upon  the  floor  1 1,  being  connected 
with  each  blind  by  a  short  iron  link.  The  ring  is  moved 
round  by  a  rack  and  spindle  which  descend  into  the  mill 
room  below,  for  the  convenience  of  the  miller.  The  mode 
of  bringing  the  sails  back  against  the  wind,  which  Mr. 
Beatson  invented,  is,  perhaps,  the  simplest  and  best  for 
that  end.  He  makes  each  sail  AI,  fig.  3,  to  consist  of  six 
or  eight  flaps  or  vanes,  AP  b  1,  b  I  c  2,  <fcc.,  moving  upon 
hinges  represented  by  the  dark  lines,  AP  b  1,  c  2,  &c.,  so 
that  the  lower  side  b  1  of  the  first  flap  wraps  over  the  hinge 
or  higher  side  of  the  second  flap,  and  so  on.  When  the 
wind,  therefore,  acts  upon  the  sail  AI,  each  flap  will  press 


WINDMILLS.  233 

upon  the  hinge  of  the  one  immediately  below  it,  and  the 
whole  surface  of  the  sail  will  he  exposed  to  its  action. 
But  when  the  sail  AI  returns  against  the  wind,  the  flaps 
will  revolve  round  upon  their  hinges,  and  present  only 
their  edjjes  to  the  wind,  as  is  represented  at  EG,  so  that 
the  resistance  occasioned  by  the  return  of  the  sail  must  be 
greatly  diminished,  and  the  motion  will  be  continued  by 
the  great  superiority  of  force  exerted  upon  the  sails  in  the 
position  AI.  In  computing  the  force  of  the  wind  upon  the 
sail  AI,  and  the  resistance  opposed  to  it  by  the  edges  of  the 
flaps  in  EG,  Mr.  Beatson  finds,  that  when  the  pressure 
upon  the  former  is  1872  pounds,  die  resistance  opposed  by 
the  latter  is  only  about  3G  pounds,  or  j2  part  of  the  whole 
force ;  but  he  neglects  the  action  of  the  wind  upon  the 
arms,  CA,  &c.,  and  the  frames  which  carry  the  sails,  be- 
cause they  expose  the  same  surface  in  the  position  AI,  as 
in  the  position  EG.  This  omission,  however,  has  a  ten- 
dency to  mislead  us  in  the  present  case,  as  we  shall  now 
see  ;  for  we  ought  to  compare  the  whole  force  exerted 
upon  the  arms,  as  well  as  the  sail,  with  the  whole  resistance 
which  these  arms  and  the  edges  of  the  flaps  oppose  to  the 
motion  of  the  windmill.  By  inspecting  the  figure  it  will 
appear,  that  if  the  force  upon  the  edges  of  the  flaps,  which 
Mr.  Beatson  supposed  to  be  12  in  number,  amounts  to  36 
pounds,  the  force  spent  upon  the  bars  CD,  DG,  GF,  FE, 
&c.,  cannot  be  less  than  60  pounds.  Now,  since  these  bara 
are  acted  upon  with  an  equal  force,  when  the  sails  have  the 
position  AI,  1872  -f  60  =  1932  will  be  the  force  exerted 
upon  the  sail  AI,  and  its  appendages,  while  the  opposite 
force  upon  the  bars  and  edges  of  the  flaps  when  returning 
against  the  wind  will  be  36  -f-  60  =  96  pounds,  which  is 
nearly  ^  of  1932,  instead  of  -j-1^  as  computed  by  Mr.  Beat 
son.  Hence  we  may  see  the  advantages  which  will  pro 
bably  arise  from  usinff  a  screen  for  the  returning  sail  instead 
of  movable  flaps,  as  it  will  preserve  not  only  the  sails,  but 
the  arms  and  the  frame  which  supports  it,  from  the  action 
of  the  wind. 

f  Figures  4  and  5,  plate  2d,  represent  the  most  improved 
form  of  the  vertical  windmill ;  aaua,  are  the  vanes  or  sails 
of  the  mill,  which  communicate  motion  to  the  wind-shaft  b 
and  the  crown  wheel  c;  f/,  the  centre  wheel  which  conveys 
this  motion  along  the  shaft  e  to  the  spur-wheel  f;  g,  a 
wheel,  or  trundle,  on  the  end  of  the  spindle  of  the  upper 
20* 


234  fx\  fcuiviA  TIL  a. 


or  turning  millstone  ;  i,  the  case  in  which  the  millstones 
are  placed  ;  k,  the  bridge-tree  which  supports  the  spindle 
of  the  turning-stone  ;  /,  another  wheel,  or  trundle,  on  the 
end  of  the  shaft  m,  which  conveys  the  motion  lower  down 
the  building  to  another  spur-wheel  n;  this  spur-wheel  puts 
other  two  millstones  in  motion  at  pleasure,  in  the  same 
manner  as  the  former  ;  o,  the  brake,  or  rubber,  for  stopping 
the  mill,  it  operates  by  friction  ;  p,  the  governor  for  regu- 
lating the  motion,  by  opening  or  shutting  the  wind-boards 
on  the  vanes  ;  q,  the  director  which  carries  round  the  roof 
with  the  wind,  by  keeping  the  vanes  always  at  right  angles 
to  it.  On  the  spindle  of  this  director  is  placed  an  endless 
screw,  .working  into  a  wheel  which  turns  a  shaft  having  a 
pinion  fixed  at  the  other  end  of  it.  This  pinion  works  into 
another  wheel  connected  with  the  rack  pinion,  which  puts 
the  whole  roof  in  motion. 

The  wind  does  not  act  perpendicularly  on  the  sails  of  a 
wind-mill,  but  at  a  certain  angle,  and  the  sail  varies  in  the 
degree  of  its  inclination  at  different  distances  from  the 
centre  of  motion,  in  resemblance  to  the  wing  of  a  bird  ; 
this  is  called  the  weathering  of  the  sail.  The  angles  of 
weathering  have  been  found  by  Smeaton  as  follows.  The 
radius  being  divided  into  6  equal  parts,  and  the  first  part 
from  the  centre  being  called  1,  the  last  6. 

Distance  from  Angle  with  Angle  with  the 

the  centre.  the  axis.  plane  of  motion. 

1  72  18 

2  71  19 

3  72  18 

4  74  16 

5  77£  12£ 

6  83  7 

Smeaton  gives  the  following  maxims  for  the  construction 
rf  windmills. 

1.  The  velocity  of  the  windmill  sails,  whether  unloaded 
or  loaded,  so  as  to  produce  a  maximum,  is  nearly  as  the 
velocity  of  the  wind,  their  shape  and  motion  being  the  same. 
2.  The  load  at  the  maximum  is  nearly  but  somewhat  less 
than,  as  the  square  of  the  velocity  of  the  wind,  the  shape 
and  position  of  the  sails  being  the  same.  3.  The  effects  of 
the  same  sails  at  a  maximum  are  nearly  but  somewhat  less 
than,  as  the  cubes  of  the  velocity  of  the  wind.  4.  The  load 


WINDMILLS. 


233 


of  the  same  sails  at  the  maximum  is  nearly  as  the  squares, 
antl  their  effects  as  the  cubes  of  their  number  of  turns  in  a 
given  time.  5.  When  the  sails  are  loaded  so  as  to  produce 
a  maximum  at  a  given  velocity,  and  the  velocity  of  the 
wind  increases,  the  load  continuing  the  same,  then,  when 
the  increase  of  the  velocity  of  the  wind  is  small,  the  effect 
will  be  nearly  as  the  squares  of  the  velocities  ;  but  when 
the  velocity  of  the  wind  is  double,  the  effects  will  be  nearly 
as  10  to  27rl ;  and  when  the  velocities  compared  are  more 
than  double  of  that  where  the  given  load  produces  a  maxi- 
mum, the  effect  increases  only  as  the  increase  of  the  velo- 
city of  the  wind.  6.  If  sails  are  of  a  similar  figure  and 
position,  the  number  of  turns  in  a  given  time  will  be  in- 
versely as  the  radius  of  length  of  the  sail.  7.  The  load  at 
a  maximum  that  sails  of  a  similar  figure  and  position  will 
overcome,  at  a  given  distance  from  the  centre  of  motion, 
will  be  as  the  cube  of  the  radius.  8.  The  effect  of  sails  of 
similar  figure  and  position  are  as  the  square  of  the  radius. 

Rules  for  modelling,  the  sails  of  Windmills. 

The  accompanying  cut  is  the 
front  view  of  one  sail  of  a  wind- 
mill. The  length  of  the  arm 
AA,  called  by  workmen  the 
whip,  is  measured  from  the 
centre  of  the  great  shaft  B,  to 
the  outermost  bar  19.  The  breadth  of  the  face  of  the  whip 
A  next  the  centre,  is  -j1^  of  the  length  of  the  whip,  and  its 
thickness  at  the  same  end  is  |  of  the  breadth ;  and  the  back 
side  is  made  parallel  to  the  face  for  half  the  length  of  the 
whip :  the  small  end  of  the  whip  is  square,  and  "at  its  end 
is  1-1 6th  of  the  length  of  the  whip,  or  half  the  breadth  at 
the  great  end. 

From  the  centre  of  the  shaft  B,  to  the  nearest  bar  1  of 
the  lattice  is  l-7th  of  the  whip,  the  remaining  space  of 
6-7 ths  of  the  whip  is  equally  divided  into  19  spaces  ;  l-9th 
of  one  of  these  spaces  gives  the  size  of  the  mortice,  which 
must  be  made  square. 

To  prepare  the  whip  for  mortising,  strike  a  gauge  score  at 
about  three-quarters  of  an  inch  from  the  face  on  each  side, 
and  the  gauge  score  on  the  leading  side,  4,  5,  will  give  the 
face  of  all  the  bars  on  each  side ;  but  on  the  other  side  the 
faces  of  all  the  bars  will  fall  deeper  than  the  gauge  score, 


236  PNEUMATICS. 

according  to  a  certain  rule,  which  is  this : — Extend  the 
compasses  to  any  distance  at  pleasure,  so  that  6  times  tlfat 
extent  may  be  greater  than  the  breadth  of  the  whip  at  the 
seventh  bar.  Set  off  these  six  spaces  upon  a  straight  line 
for  a  base,  at  the  end  of  which  raise  a  perpendicular ;  set 
off  the  same  six  spaces  on  the  perpendicular,  and  divide  the 
two  spaces  on  the  perpendicular  which  are  farthest  from 
the  base,  each  into  6  equal  parts,  so  that  these  two  spaces 
will  contain  13  points.  Join  each  of  these  13  points  with 
the  end  of  the  base  farthest  from  the  perpendicular. 

To  apply  this  scale  to  any  given  case,  set  off  the  breadth 
of  the  whip  at  the  last  bar  (that  is  the  bar  at  the  extremity 
of  the  sail)  from  the  centre  of  the  scale,  along  the  base  to- 
wards the  perpendicular,  and  at  this  point  raise  a  perpen- 
dicular to  cut  the  oblique  line  nearest  the  base ;  also  set  off 
the  breadth  at  the  seventh  bar  in  the  same  manner,  and  at 
this  point  raise  a  perpendicular  to  cut  off  the  thirteenth 
oblique  line.  Now,  from  the  point  where  the  first  of  these 
two  perpendiculars  cuts  the  first  oblique  line  from  the  base, 
to  the  intersection  of  the  second  perpendicular  with  the 
thirteenth  oblique  line,  there  is  drawn  a  line  joining  the 
two  points  of  intersection  ;  and  perpendiculars  being  drawn 
from  the  points  where  this  joining  line  cuts  the  oblique 
lines  to  the  base,  will  be  the  several  distances  of  the  face 
of  each  bar  from  the  gauge  line.  These  distances  give  a 
difference,  set  off  for  each  bar  to  the  seventh,  which  must 
be  set  off  for  all  the  rest  to  the  first.  The  length  of  the 
longest  bar  is  -|  of  the  whip. 

We  now  proceed  to  show  the  method  of  weathering  the 
sails.  Draw  AB  =  the  length  of  the  vane,  BC  its  breadth, 
and  BCD  the  angle  of  the  weather  at  the  extremity  of  the 
vane,  equal  to  20  degrees.  With  the  length  of  the  vane 
AB,  and  breadth  BC,  construct  the  isosceles  triangle  ABC  ; 
and  from  the  point  B,  draw  BD  perpendicular  *,o  CB,  then 
BD  is  the  proper  depth  of  the  vane. 

Divide  the  line  AB  into  any  number  of  parts,  say  four 
at  these  divisions  draw  the  lines  1  E,  2  F,  3  G,  &c 


WINDMILLS.  237 

parallel  to  the  line  BC.  Also,  from  the  points  of  division, 
1,  2,  3,  &c.,  draw  the  lines  1  i,  2  k,  3  1,  &c.  perpendicular 
to  1  E,  2  F,  3  G,  &c.,  all  of  them  equal  in  length  to  BD. 
Join  Ei,  Fk,  Gl,  &c.,  then  the  angles  1  Ei,  2  Fk,  3  Gl, 
&c.,  are  the  angles  of  the  weather  for  these  divisions  of  the 
vane  ;  and  if  the  triangles  be  conceived  to  stand  perpen- 
dicular to  the  paper,  the  angles  i,  k,  1,  and  D,  denoting  the 
vertical  angles,  the  hypotenuses  of  these  triangles  will 
give  a  perfect  idea  of  the  weathering  of  the  vane  as  it  recedes 
from  the  centre. 


HEAT,   STEAM,  &c, 


IT  would  be  out  of  place  in  a  work  of  this  nature  to 
enter  into  a  minute  detail  respecting  the  nature  of  heat ;  in 
this  section,  therefore,  we  shall  confine  ourselves  to  a  de- 
scription of  the  more  important  of  its  mechanical  properties. 

Heat  expands  bodies,  that  is,  increases  their  dimensions. 
Different  bodies  expand  differently  by  the  application  of 
the  same  quantity  of  heat.  With  the  same  degree  of  heat, 
solids  expand  less  than  liquids,  and  liquids  less  than  gases. 

On  the  principle  that  bodies  expand  by  heat,  is  con- 
structed the  Thermometer.  The  action  of  this  instrument 
is  very  simple.  It  consists  of  a  small  glass  tube  with  a 
hollow  bulb  at  one  end,  and  at  the  other  end  it  is  closed. 
The  bulb  is  filled  with  mercury,  as  likewise  a  part  of  the 
tube,  the  other  portion  of  the  tube  being  entirely  deprived 
of  air.  When  heat  is  applied  to  the  bulb  of  the  thermome- 
ter, the  mercury  expands  and  rises  in  the  tube,  and  accord- 
ing to  the  degree  of  heat  applied  to  it,  so  will  the  mercury 
rise.  To  the  tube  there  is  attached  a  divided  scale,  to  de- 
note the  degrees  of  heat  by  the  rising  of  the  mercury,  which 
scale  is  thus  formed.  The  bulb  of  the  thermometer  is  put 
into 'melting  ice,  and  the  height  of  the  mercury  is  marked 
om  the  scale ;  this  is  called  the  freezing  point,  and  num- 
bered 32.  The  bulb  is  then  put  into  boiling  water,  and  the 
height  of  the  mercufy  in  the  tube  is  marked  upon  the  scale 
and  numbered  212 — this  is  called  the  boiling  point.  The 
space  betwixt  these  two  points  on  the  scale  is  divided  into 
180  equal  parts,  called  degrees,  and  the  scale  is  then  ex- 
tended both  above  and  below  these  points.  This  is  the 
scale  commonly  used  in  this  country,  and  is  known  by  the 
name  of  its  inventor,  Fahrenheit.  But  the  French  and  many 
philosophers  in  Britain  use  a  thermometer  having  a  scale 
of  much  more  simple  construction,  called,  from  the  nature 
of  its  divisions,  the  Centigrade  scale.  The  freezing  point, 
which  in  Fahrenheit  is  marked  32,  is  in  the  Centigrade 

238 


HEAT.  239 

marked  0  or  zero  ;  and  the  boiling  point,  in  Fahrenheit 
marked  21  2,  is  in  the  Centigrade  marked  100.  In  Reaumur's 
thermometer  the  freezing  point  is  murked  0,  and  the  boiling 
point  80. 

Let  F  represent  Fahrenheit,  R  Reaumur,  and  C  Centi- 
grade, then  we  have  the  following  rules  for  converting  the 
degrees  of  any  one  of  these  thermometers  into  the  corres- 
ponding temperature,  as  marked  in  the  others  :  — 

(1.)  F  =  C  x  1-8  -f  32. 

q  ft 
(2.)  F  =—  +  32. 


(5.)  R  _ 


(6.)  R  =  C  x  0-8. 

Thus  185  Fahrenheit's  will  be  found  to  correspond  to 
85  of  the  Centigrade,  and  68  of  Reaumur's  thermometer. 
(1.)  85  x  1-8  +  32  =  185. 

(2.)  !L^  +  32  =  185. 

(3.)  "^  _  85. 

(40    •         £-  85. 

(5.)  *  X  (185  -32)  =  68. 

(6.)  85  X  0-8  =  68. 

,  'There  are  many  other  particulars  regarding  the  thermo- 
meter which  it  would  be  inconsistent  with  the  design  of 
these  pages  to  consider  :  what  we  have  said  will  be  sufficient 
forxhe  understanding  of  what  is  hereafter  to  follow  on  the 
suoject  of  steam,  &c. 

Before  we  introduced  the  subject  of  the  thermometer,  we 
stated  the  fact  of  the  expansion  of  bodies  by  heat.  Bars 
of  the  following  substances,  whose  length  at  'a  temperature 


240  HEAT. 

of  32  was  1,  were  heated  to  212  Fahrenheit,  and  expanded 
so  as  to  become, 


Cast  iron, 1-00110940 

Steel, 1-00118990 


Copper, 1-00191880 

Brass, 1-00188971 


This  is  the  expansion  in  length  ;  the  expansion  in  length, 
breadth,  and  thickness,  will  be  found  by  multiplying  the 
above  numbers  by  3. 

The  effects  of  different  degrees  of  heat  on  different  bodies, 
according  to  Fahrenheit's  scale,  are  shown  below. 

Cast  iron  thoroughly  melted, 20577 

Cast  iron  begins  to  melt, 17977 

Greatest  heat  of  a  common  smith's  forge, 17327 

Flint  glass  furnace,  strongest  heat, 1 5897 

Welding  heat  of  iron,  (greatest) 13427 

Swedish  copper  melts, 4587 

Brass  melts, 3807 

Iron  red  hot  in  the  twilight, 884 

Heat  of  a  common  fire, 790 

Iron  bright  red  in  the  dark, 752 

Zinc  melts, 700 

Mercury  boils, 672 

Lead  melts, 594 

The  surface  of  polished  steel  becomes  uniformly 

deep  blue, 580 

The   surface  of  polished  steel  becomes  a  pale 

straw  colour, 460 

Tin  melts, 442 

A  mixture  of  3  tin  and  2  lead  melts, 332 

Hgat  passes  through  different  bodies  with  very  different 
degrees  of  velocity,  and  according  to  the  rapidity  or  slow- 
ness with  which  heat  passes  through  any  body,  it  is  said  to 
be  a  good  or  a  bad  conductor  of  heat.  The  conducting 
power  of  copper  being  1,  that  of  brass  will  be  1,  iron,  1-1, 
tin,  1-7,  lead,  2-5.  The  densest  bodies  are  generally  the 
'best  conductors  of  heat ;  but  this  is  not  ufijy.ersal,  as  platina, 
one  of  the  densest  of  all  metals,  is  one  of  the  worst  con- 
ductors. Earthy  substances  are  much  inferior  to  metals  in 
their  conducting  power,  and  the  worst  conductors  of  all  are 
the  coverings  of  animals. 

When  heated  bodies  are  exposed  to  the  air  they  lose  por- 
tions of  their  heat  by  projection  in  right  lines  into  space 
from  all  parts  of  their  surface.  This  is  called  the  radiatior 


I 


HEAT.  241 

of  heat.  Bodies  which  radiate  heat  best  have  the  power  of 
absorbing  it  in  the  same  proportion,  and  the  least  power 
of  reflecting  it ;  hence,  in  leading  steam  through  a  room, 
it  would  be  absurd  to  use  black  pipes,  because,  in  that  case, 
much  of  the  heat  would  escape  by  radiation  before  the 
steam- would  be  carried  to  the  place  where  it  was  to  be  used. 
If  the  steam  is  used  to  heat  the  apartment,  black  pipes  are 
the  best.  Hence  the  cylinder  of  a  steam  engine  ought  to 
be  polished,  but  the  condenser  should  not.  Vessels  in- 
tended to  receive  heat  should  be  black. 

The  comparative  quantities  of  heat  existing  in  different 
bodies  may  be  ascertained  by  marking  the  time  which  equal 
quantities  of  them  require  to  cool  a  certain  number  of  de- 
grees, reckoning  their  capacities  for  heat  to  be  as  these 
times  estimated  by  the  volume ;  or,  if  divided  by  the  spe- 
cific gravity  of  the  substance,  by  the  weight. 

It  is  necessary  here  to  distinguish  carefully  between  what 
is  called  the  specific  heat  of  a  body,  and  its  capacity  for  heat, 
these  two  terms  being  often  confounded.  If  we  take  two 
bodies  at  the  same  temperature,  and  expose  them  to  the 
action  of  a  greater  heat,  it  will  be  found  that  one  body 
will  have  absorbed  a  greater  quantity  of  heat  than  the 
other,  by  the  time  that  they  have  acquired  an  equal  tern 
perature  ;  and  the  amount  of  this  additional  heat,  referred 
to  some  standard,  is  denominated  the  specific  heat  of  the 
body.  Thus  if  it  be  found  that  it  requires  1  degree  of  heat 
to  raise  water  from  one  temperature,  T,  to  another  tempera- 
ture, f^and  if  to  produce  the  same  change  of  temperature 
in  steam-it  requires  0-847  degrees,  then  is  0-847  the  specific 
heat  of  steam,  water,  as  the  standard,  being  1-000.  The 
capacity  of  one  body  for  heat  compared  to  another  is  not 
the  relative  quantities  of  heat  required  to  raise  them  a 
certain  number  of  degrees,  but  the  absolute  quantities  con- 
tained in  them  at  the  same  temperature. 

CAPACITIES  OF   BODIES   FOR  HEAT. 

GASES. 

Atmospheric  air, 1  -7900 

Aqueous  vapour 1-550C 

Carbonic  acid  gas, .'.1-0454 

LIQUIDS. 

Alcohol, 1-0860 

W/iter, 1-0000 

21 


842 


HEAT. 


Solution  of  muriate  of  soda,  1  in  10  of  water, -9360 

Sulphjric  acid,  diluted  with  10  parts  water, -9250 

Solution  of  muriate  of  soda  in  6'4  of  water, -9050 

Olive  oil, t -7100 

Nitric  acid,  specific  gravity  1-29895, .-6613 

Sulphuric  acid,  with  an  equal  weight  of  water, -6050 

Nitrous  acid,  specific  gravity  1  -354, -5760 

Linseed  oil, -5280 

Oil  of  turpentine, '4720 

Quicksilver,  specific  gravity  13-30, -0330 

SOLIDS. 

Ice, , -9000 

White  wax, -4500 

Quicklime,  with  water,  in  the  proportion  of  16  to  9,  -4391 

Quicklime, -3000 

Quicklime  saturated  with  water,  and  dried, -2800 

Pit  coal -2800 

Pit  coal, -2777 

Rust  of  iron, -2500 

Flint  glass,  specific  gravity  287, -1900 

Iron, -1300 

Hardened  steel, -1230 

Soft  bar  iron,  specific  gravity  7'724, -1190 

Brass,  specific  gravity  8-356, -1160 

Copper,  specific  gravity  8-785, '1140 

Sheet  iron, «-1099 

Zinc,  specific  gravity  8-154, -1020 

White  lead, -0670 

Lead, -0352 

Specific  heats.  Specific  heats. 

Specific  heat  of  water  equal  1.  Specific  heat  of  water  equal  1. 

Bismuth, 0-0288  Tellurium, 0-09 12 

Lead, -0-0293  Copper, 0-0949 

Gold, 0-0298  Nickel, 0-1035 

Platinum,.- 0-0314   |  Iron 0-1100 

Tin 0-0514  Cobalt, 0-1498 

Silver, 0-0557  Sulphur, 0-1880 

Zinc, 0-0927 


HEAT.  243 

Large  quantities  of  heat  must  enter  into  bodies,  and  be 
concealed,  to  enable  them  to  pass  from  the  solid  to  the  fluid 
state,  or  from  the  fluid  state  to  that  of  vapour.  Thus  the 
quantity  of  heat  necessary  to  convert  any  given  weight  of 
ice  into  water*  would  raise  the  same  weight  of  water  14( 
degrees  of  Fahrenheit.  This  quantity  of  heat  is  not  sensi 
ble,  but  is,  as  it  were,  kept  hid  or  laltn!  :  nor  can  it  be  de 
tected  by  the  touch,  or  by  application  <if  the  thermometer. 

Every  addition  of  heat  applied  to  water  in  a  fluid  state, 
raises  the  temperature  until  it  arrives  at  the  boiling  point ; 
but  however  violently  the  fluid  may  boil,  it  does  not  become 
hotter,  nor  does  the  steam  that  arises  from  it  indicate  a 
greater  degree  of  heat  than  the  water:  hence,  <i  large  pro- 
portion of  the  heat  must  enter  into  the  steam  and  become 
latent.  The  quantity  of  heat  that  becomes  latent  in  steam, 
was  found  by  Dr.  Black  to  be  810  decrees  of  Fahrenheit. 

Under  the  common  pressure  of  the  atmosphere  at  the 
surface  of  the  earth,  (15  Ibs.  on  the  square  inch,)  water 
cannot  be  raised  above  a  temperature  of  212  Fahr.  :  but 
when  exposed  to  greater  pressure,  by  being  confined  in  a 
vessel,  the  water  may  be  raised  to  a  much  higher  degree  of 
heat,  and  if,  in  this  state  of  confinement,  the  heat  applied 
be  insufficient  to  cause  the  water  to  boil  :  if  the  vessel  should 
be  open,  steam  will  rush  out,  and  the  water  which  remains 
will  fall  in  temperature  to  212.  On  the  contrary,  water 
boils  at  very  low  temperatures  when  the  pressure  is  dimi- 
nished ;  as  in  an  exhausted  receiver,  nr  at  the  tops  of 
mountains. 

When  the  temperature  of  steam  is  reduced,  it  assumes 
again  the  fluid  form,  and  the  quantity  of  latent  heat  set  free 
by  steam  in  passing  to  the  state  of  water,  has  been  found, 
by  Mr.  Watt,  to.be  945  degrees.  He  also  found  that  a 
cubic  inch  of  water  may  be  converted  into  a  cubic  foot  of 
steam  ;  and  that  when  this  steam  is  condensed,  by  injecting 
cold  water,  tlie  latent  heat  which  the  steam  gives  out  in 
passing  to  the  fluid  state,  would  be  sufficient  to  heal  6  cubic 
inches  of  water  to  the  temperature  of  212,  or  the  boiling 
point.  It  is  generally  considered  that  steam  raised  from 
•boiling  water  occupies  18  hundred  times  as  much  space  as 
the  water  did  from  which  it  was  raised,  and  instead  of 
making  the  latent  heat  of  steam  810,  as  Dr.  Black  found  it, 
mjore  correct  experiments  show  it  to  be  1000,  at  the  ccm 
nion  pressures  of  the  atmosphere ;  but  the  latent  heat  of 


244  HEAT. 

steam  is  inversely  proportional  to  the  degree  of  pressure 
under  which  it  is  produced ;  that  is,  the  latent  heat  is 
greatest  where  the  pressure  is  least,  and  least  where  the 
pressure  is  greatest. 

It  has  lately  been  discovered  that  the  sensible  heat  and 
latent  heat  of  steam  at  any  one  temperature  added  together, 
give  a  sum  which  is  constant ;  that  is  to  say,  which  is  the 
sum  of  the  sensible  and  latent  heat  of  any  other  tempera- 
ture, or  under  any  other  pressure.  Now,  the  sensible  heat 
of  steam  at  the  ordinary  pressure  of  the  atmosphere  is 
212  —  32  =  180  ;  and  the  latent  heat  has  been  found  to  be 
1000,  their  sum  is  1180,  which  is  the  constant  sum  of  the 
latent  and  sensib'e  heats  of  steam  under  any  other  pressure. 
Thus,  at  the  temperature  of  248,  where  the  elastic  force  of 
the  steam  is  equal  to  two  atmospheres,  or  a  pressure  of  30 
Ibs.  on  the  square  inch,  the  sensible  heat  will  be  248  —  32 
=  216,  wherefore  the  latent  heat  is  1180  —  216  =  964, 
and  so  of  the  other  temperatures. 

It  has  also  been  found  that  while  the  elasticity  of  steam 
increases  in  geometrical  progression,  with  a  ratio  of  2,  the 
latent  heat  diminishes  with  a  ratio  of  T0306,  differing  hot 
very  materially  from  a  unit. 

Many  experiments  have  been  made  to  ascertain  the 
elastic  force  of  steam  of  various  temperatures.  The  most 
valuable  of  them  are  those  recently  made  by  the  French 
academicians,  the  results  of  which  are  given  below  in  a 
tabular  form  ;  and  the  practical  man  will  duly  estimate  the 
value  of  this  gift  of  science. 

The  following  simple  rule  is  easily  remembered  and  ap- 
plied, and  comes  near  enough  to  the  truth  for  all  practical 
uses. 

/temperature  +  lOOV 

)  =  the  force  of  the 
v  177 

steam  in  inches  of  mercury.  Thus  if  the  temperature  be 
307, then, 

307  +  100      _. 

~T77~ 

then  2-3  X  2-3  x  2-3  x  2-3  x  2-3  X  2-3  =  148-0359,  thin 
divrJed  by  30,  g\ves  the  atmospheres, 

148-0359 

— —  =  4'93  atmospheres. 


HEAT 


245 


TABLE  OF  THE  ELASTICITY  OF  STEAM, 

BY  M.  ARAGO  AND  OTHERS. 


Elasticity  of 

•Irani,    the 
piw.  at    the 

a!ni'i-|.!irrc 
being  1. 

temp,  in  de*.  of 
Fahrenheit. 

Elasticity  of 
•team,    the 

in--,    "f    'In; 
atnirwphere 

being  1. 

Corresponding 
temp,   in    deg,   of 
Fahrenheit. 

1 

212 

13 

380-66 

u 

231 

14 

386-94 

2 

250-5 

15 

392-86 

2£ 

263-8 

16 

398-48 

3 

275-2 

17 

403-83 

3d 

285 

18 

408-92 

4 

293-7 

19 

413-78 

4d 

300-3 

20 

418-46 

5 

307-5 

21 

422-96 

5£ 

314-24 

22 

427-25 

6 

320-30 

23 

431-42 

6d 

326-26 

24 

435-56 

7 

331-7 

25 

439-34 

71 

336-86 

30 

457-16 

8 

341-78 

35 

472-73 

9 

350-78 

40 

486-59 

10 

358-78 

45 

499-24 

11 

366-85 

50 

510-6 

12 

374 

in  constructing  this  table  the  results  were  derived  from 
experiments  up  to  24  atmospheres,  after  which  the  formula 
which  follows  was  employed. 

E  =  (l  +T  +  0-7153)  5 

Where  E  represents  the  elasticity,  and  T  the  temperature, 
by  the  centigrade  thermometer,  regarding  100°  as  unity, 
and  T  the  excess  of  tepiperature  above  100°  It  maybe 
observed  that  this  formula  is  more  accurate  in  veiy  high 
temperatures  than  for  low. 


21* 


246 


HEAT. 


ELASTIC  FORCE  OF  STEAM,  BY  DR.  URE. 


_       F.lasHr  II  _ 

Elastic 

Elastic 

Elastic 

Temp. 

force,  j  lemP- 

Temp. 

force 

Temp. 

force. 

24° 

0-170 

155° 

8-500 

242  3 

53-600 

281-8° 

104-400 

32 

0-200 

160 

9-600  245 

56-340 

283-8 

107-700 

40 

0-250 

165 

10-800  245-8 

57-100 

285-2 

112-200 

50 

0-360 

170 

12-050  218-5 

60-400 

287-2 

114-800 

55 

0-416! 

175 

13-550  250 

61-900 

289 

118-200 

60 

0-516 

180 

15-160 

251-6 

63-500 

290 

120-150 

65 

0-630JIIS5 

16-900 

254-5 

66-700 

292-3 

123-100 

70 

0-726 

190 

19*900 

255 

67-250 

294 

126-700 

75 

0-860' 

195 

21-100 

257-5 

69-800 

295 

129-000 

80 

1-010! 

200 

23-600 

260 

72-300 

295-6 

130-400 

85 

1-170! 

205 

25-900 

260-4 

72-800 

297-1 

133-900 

90 

1-360' 

210 

28-830 

262-8 

75-900 

298-8 

137-400 

95 

1-640 

212 

30-000 

264-9 

77-900 

300 

139-700 

100 

1-860! 

216-6 

33-400 

265 

78-040 

300-6 

140-900 

105 

2-100! 

220 

35-540 

267 

81-900 

302 

144-300 

110  12-456 

221-6 

36-700 

269 

84-900 

303-8 

147-700 

115 

2-820 

225 

39-110 

270 

86-300 

305 

150-560 

120 

3-300! 

226-3 

40-100 

271-2 

88-000 

306-8 

155-400 

125  (3-830 

230 

43-100 

273-7 

91-200 

308 

157-700 

130  4-366: 

230-5  43-500 

275 

93-480 

310 

161-300 

135  5-070 

234-5 

46-800 

275-7 

94-600 

311-4 

164-800 

140  i5-770; 

235 

47-220 

277-9 

97-800 

312 

167-000 

145 

6-600 

238-5  50-300 

279-5 

101-600 

312 

165-5 

150 

7-530 

240  51-700J280 

101-900 

Before  we  describe  the  application  of  steam  in  the 
steam  engine,  we  shall  briefly  allude  to  some  other  useful 
purposes  to  which  it  has  been  subjected.  It  has  been  as- 
certained that  one  cubic  foot  of  boiler  will  heat  about  2000 
feet  of  space,  in  a  cotton  mill,  to  an  average  heat  of  about 
70°  or  80°  Fahr.  It  has  also  been  proved  that  one  square 
foot  of  surface  of  steam  pipe  is  adequate  to  the  warming  of 
200  cubic  feet  of  space.  This  quantity  is  adapted  to  a  well 
finished,  ordinary  brick  or  stone  building.  Cast  iron  pipes 
are  preferable  to  all  others  for  the  diffusion  of  heat,  the  pipes 
being  distributed  within  a  few  inches  of  the  floor.  Steam 
is  also  used  extensively  for  drying  muslin  and  calicoes. 
Large  cylinders  are  filled  with  it,  which,  diffusing  in  the 
apartment  a  temperature  of  100°  or  130°,  rapidly  dry  the 
suspended  cloth.  Experience  has  shown  that  bright  dyec1 
yarns,  like  scarlet,  dried  in  a  common  stove  heat  of  128°, 
have  their  colour  darkened,  and  acquire  a  harsh  feel ; 
while  similar  hanks,  laid  on  a  steam  pipe  heated  up  to  165° 


HEAT. 


247 


retain  the  shade  and  lustre  they  possessed  in  the  moist  state. 
Besides,  the  people  who  work  in  steam  drying  rooms  are 
healthy,  while  those  who  were  formerly  employed  in  he 
stove  heated  apartments,  became,  in  a  short  time,  sickly 
and  emaciated.  The  heating,  by  steam,  of  lar<_re  quantities 
of  water  or  otheY  liquids,  either  for  baths  or  manufactures, 
may  be  effected  in  two  ways  :  The  steam  pipe  may  be 
plunged,  with  an  open  end,  into  the  water  cistern  ;  or  the 
steam  may  be  diffused  around  the  liquid  in  the  interval  be- 
tween the  wooden  vessel  and  the  interior  metallic  case. 

Elastic  force  of  vapour  of  alcohol  of  a  specific  gravity  of 
0-813,  water  being  1. 


Alcohol  of  S.  G.  0-813. 

Temp. 

Force  of  vap. 

Temp. 

Force  of  yap. 

3-3° 

0-40 

180-0 

34-73 

40-0 

0-56 

18-2-3 

36-40 

45-0 

0-70 

185-3 

39-90 

50-0 

0-86 

190-0 

43-20 

55-0 

I'OO 

193-3 

46-60 

60-0 

1-23 

196-3 

50-10 

65-0 

1-49 

200- 

53-00 

70-0 

1-7(1 

20(5-0 

60-10 

75-0 

2-10 

210-0 

65-00 

80-0 

2-45 

214-0 

69-36 

85-0 

2-93 

21»i-0 

72-20 

90-0 

3-40 

220-0 

78-50 

95-0 

3-90 

225-0 

87-50 

100-0 

4-50 

230-0 

94-10 

105-0 

5-20 

232-0 

97-10 

110-0 

6-00 

236-0 

103-60 

115-0 

7-10 

233-0 

106-90 

1-20-0 

8-10 

210-0 

111-24 

1-25-0 

9-25 

214- 

118-20 

130-0 

10-60 

247-0 

122-10 

135-0 

12-15 

248-0 

126-10 

140-0 

-    13-90 

249-7 

131-40 

145-0 

15-95 

250-0 

132-30 

150-0 

18-00 

252-0          138-60 

155-0 

20-30 

254-3          143-70 

160-0 

22-60 

258-6          151-60 

165-0 

25-40 

260-0          155-20 

170-0 

28-30 

262-0          161-40 

173-0 

30-00 

264-0 

166-10 

178-3 

33-50 

248  STEAM    ENGINE. 


THE    STEAM    ENGINE. 

IT  is  not  consistent  with  the  plan  of  this  book,  that  we 
should  enter  into  minute  details  as  to  all  the  modifications 
and  departments  of  the  steam  engine?  a  subject  which 
would  of  itself  occupy  a  large  volume.  We  shall,  however, 
attempt  to  explain  the  leading  principles  on  which  this  in- 
valuable machine  operates,  so  that  the  mode  of  calculating 
its  effects  may  be  the  more  clearly  comprehended. 

The  engine  of  Newcomen  consists  of  a  hollow  cylinder 
furnished  with  a  solid  piston.  This  piston  is  attached 
to  a  rod,  the  top  of  which  is  connected  with  a  large  beam, 
resting  upon  a  fulcrum  in  the  centre.  To  the  other  end 
of  this  large  beam,  called  the  working  beam,  the  pump  rod 
is  attached.  When  steam  is  admitted  into  the  bottom  of 
the  cylinder,  it  will,  by  the  superiority  of  its  elastic  force 
above  the  pressure  of  the  atmosphere,  assisted  by  the  coun- 
teraction of  the  weight  of  the  pump  rod,  cause  the  piston 
to  rise  to  the  top  of  the  cylinder.  But  when  the  piston 
arrives  at  this  point,  cold  water  is  injected  into  the  cylinder, 
by  which  the  steam  is  condensed,  and  a  vacuum  formed, 
then  the  pressure  of  the  air  on  the  top  of  the  piston  will 
cause  it  to  descend  to  the  bottom  of  the  cylinder.  The 
steam  is  again  injected  and  again  condensed,  and  thus  the 
operation  of  the  machine  is  continued.  This  is  called  the 
atmospheric  engine.  It  is  liable  to  this  objection,  that 
there  is  a  great  waste  of  steam,  and  consequently  of  fuel 
incurred  in  consequence  of  the  steam  being  condensed  in 
the  cylinder,  since  the  cylinder  must  be  heated  to  a  certain 
temperature,  before  the  steam  which  it  contains  can  exert 
a  sufficient  elastic  force,  and  the  admission  of  cold  water 
cooling  it  down  below  this  temperature,  a  considerable 
quantity  of  steam  is  employed  in  again  raising  its  heat  to 
the  proper  point. 

In  order  to  obviate  this  defect,  the*  illustrious  WATT 
made  such  arrangements  as  enabled  him  to  condense  the 
steam  in  a  separate  vessel,  and  thus  to  maintain  a  uniform 
temperature  in  the  cylinder.  By  this  great  improvement 
the  effect  of  the  same  quantity  of  steam  was  increased  in 
about  the  proportion  of  12  to  7.  Such  was  the  principle 
of  Watt's  single-acting  engine  ; — but  he  afterwards  so  ar- 
ranged the  structure  of  the  machine  as  to  admit  the  steam 


STKAM    ENGINE.  249 

alternately  above  and  below  the  piston,  and  still  to  con- 
dense it  in  a  separate  vessel,  as  will  be  understood  from  the 
description  of  the  engraving,  plate  III,  which  will  be  given 
a  little  farther  on.  This  form  of  the  steam  engine  is  called 
the  double-acting  line-pressure  engine. 

The  steam  engine  was  further  improved  by  Mr.  Watt, 
by  his  shutting  off  the  steam  when  the  piston  had  passed 
through  a  portion  of  its  stroke,  by  which  means  the  acce- 
lerated motion  of  the  piston  is  counteracted,  from  the  elastic 
force  of  the  steam  diminishing:  during  its  expansion.  This 
is  the  principle  of  what  is  called  the  expansive  engine. 

In  the  kigk-presfttre  steam  engine,  the  steam,  of  high 
temperature,  is  admitted  into  the  cylinder  alternately  above 
and  below  the  piston;  but  instead  of  being  condensed,  H  is 
allowed  to  escape  into  the  atmosphere.  In  this  engine, 
which  is  the  most  simple  in  its  construction,  the  steam  acts 
by  its  elastic  force  alone. 

The  construction  of  the  low-pressure  double-acting  steam 
engine,  will  be  understood  in  its  more  minute  details,  from 
the  following  description. 

Plate  III  is  a  side  elevation  of  a  low-pressure  portable 
double-acting  steam  enginfe,  in  which  the  boiler  and  the 
other  principal  parts  are  drawn  in  section. 

After  the  flame  frflm  the  furnace  A  passes  under  the 
whole  bottom  surface  of  the  boiler,  it  enters  the  flue  C, 
from  which  it  escapes  into  a  flue  running  up  one  side  of  the 
boiler ;  from  this  side  flue  it  passes  into  the  end  flue  D, 
which  carries  it  into  a  flue  running  along  the  other  side  of 
the  boiler ;  and  from  this  last  the  smoke  is  conducted  into 
the  chimney  E.  The  bridge  B  helps  to  spread  the  flame 
over  the  bottom  of  the  boiler.  When  the  furnace  is  cleaned, 
the  plate  between  the  end  of  the  furnace  bars  and  the  bridge 
can  be  drawn  forward  by  means  of  two  handles,  (one  of 
which  only  is  shown,)  in  order  that  the  .cinders  may  be 
pushed  over  the  end  of  the  furnace  bars  into  the  ashpit. 

If  one  of  the  gauge  cocks,  FF,  is  opened,  it  will  emit 

steam ;  and  the  other  cock  if  opened  will  blow  out  water 

.  if  the  boiler  be  just  as  full  of  water  as  it  ought  to  be.     As 

•    these  cocks  stop  up  sometimes,  a  wire  may  be  passed  down 

through   them,  if  the  part  above  the  key  is  not  bent  over. 

^The  writer  should  always  stand  somewhere  between   the 

'dotted   lines   passing  below  the  ends  of  the  gauge  cocks. 

,  <"•»  is  a  small  valve  opening  inwards,  placed  in  the  man-hole 


250  STEAM    ENGINE. 

i 

door,  to  keep  the  sides  of  the  boiler  from  being  pressed 
together  by  the  force  of  the  atmosphere,  if  the  steam  should 
happen  to  be  suddenly  condensed  by  the  water  that  feeds 
the  boiler.  HH  is  the  feed  pipe,  and  the  small  valve  sus- 
pended from  the  point  O,  of  the  lever  K,  regulates  the 
quantity  of  water  passing  into  the  boiler;  the  lever  which 
works  the  feed  valve  is  connected  by  means  of  a  rod  to  the 
float  I,  which  rises  or  falls  along  with  the  water  in  the 
boiler,  and  this  opens  or  shuts  the  valve,  according  as  the 
water  stands  low  or  high  in  the  boiler.  The  pipe  L  con- 
ducts the  water  into  the  feed  pipe  from  a  cistern  fixed 
above  the  boiler  house,  which  is  kept  full  by  means  of  the 
hot  water  pump,  which  takes  in  water  from  the  hot  well. 
The  cistern  on  the  top  of  the  boiler  house  should  be  large 
enough  to  fill  the  boiler,  as  also  the  large  cistern  on  which 
the  engine  stands,  if  they  should  happen  to  be  empty  at  any 
time.  The  pipe  M  carries  away  any  overplus  water  from 
the  feed  pipe.  NN  is  the  pipe  which  conveys  the  steam 
from  the  boiler  into  the  nozles,  and  the  safety  valve  is 
placed  above  the  bend  in  it.  Q  is  a  section  of  the  cylinder, 
showing  also  the  outside  of  the  metallic  piston.  The  oblong 
opening,  near  the  top  of  the  condenser  R,  admits  a  jet  of 
cold  water  to  condense  the  steam  after  it  has  acted  in  the 
cylinder.  The  injection  cock  is  boiled  to  the  outside  of 
the  oblong  opening,  and  the  water  which  is  forced  through 
it  into  the  condenser  by  the  pressure  of  the  atmosphere  is 
taken  from  the  large  cistern  on  which  the  engine  stands ; 
this  cistern  is  always  kept  nearly  full  by  the  cold  water 
pump.  The  hot  and  cold  water  pumps  are  both  wrought 
off  the  same  spindle  P,  fixed  in  the  working  beam,  a  pump 
being  attached  to  each  end  of  the  spindle.  The  foot  valve 
S  is  placed  between  the  condenser  R,  and  the  air  pump  T. 
The  bucket  shown  in  the  air  pump  is  not  sectioned.  The 
valve  in  the  air  pump  bucket,  and  the  discharging  valve, 
which  opens  into  the  hot  well  on  the  top  of  the  air  pump, 
have  each  a  shallow  flat-bottomed  recess  turned  on  the  top, 
so  as  to  fit  nicely  the  flat-bottomed  disks  X  and  W  ;  the 
one  disk  is  keyed  on  the  air  pump  rod,  and  the  other  is 
fixed  by  means  of  studs  and  nuts  to  khe  hot  well ;  as  it 
gives  more  water  way,  it  is  an  improvement  to  have  the 
recess  in  the  valve,  rather  than  in  the  disk.  If  each  valve 
had  not  a  recess  turned  in  it  to  contain  a  quantity  of  water, 
which,  as  it  is  forced  out  by  the  disk,  reduces  the  momen- 


STEAM    ENGINE.  251 

turn  of  the  valve  by  degrees  ;  the  stroke  of  the  valve  on  its 
disk  or  guard  would  be  very  great,  and  the  parts  would 
soon  work  out  of  order.  The  pipe  which  carries  away  the 
water  that  is  pumped  out  of  the  condenser  by  the  air  pump, 
is  shown  near  the  top  of  the  hot  well,  on  the  side  farthest 
from  the  cylinder. 

The  way  in  which  the  tire  is  regulated,  is  as  follows  : — 
When  the  steam  gets  too  strong,  the  water  in  the  boiler 
rises  in  the  feed  pipe,  and  carries  up  the  float  W ;  and  as 
the  float  is  connected  by  a  chain  and  a  pulley  with  the 
damper  V,  the  damper  descends  into  the  flu*e,  and  reduces 
the  draught  in  the  furnace,  and  the  force  of  the  steam. 
Again,  if  the  steam  gets  too  low,  the  float  falls  and  raises 
the  damper,  to  increase  the  draught.  The  two  pulleys 
which  form  the  connexion  between  the  damper  and  the 
float  are  both  fixed  on  one  shaft ;  on  atcount  of  the  one 
being  placed  exactly  behind  the  other,  one  of  them  only 
can  be  seen. 

As  the  balls  YY  are  carried  round  along  with  the  rod  Z, 
when  the  engine  is  going  too  quick,  the  balls  by  their  cen- 
trifugal force  fly  out,  and  the  rods  and  levers  in  connexion 
with  them  shut  more  or  less  a  valve  at  A',  in  the  steam 
pipe ;  if  the  engine  goes  too  slow,  the  balls  fall  down,  and 
open  this  valve  to  give  the  engine  more  steam  to  bring  up 
its  motion.  The  rod  B',  and  the  lever  C',  form  part  of  the 
connexion  with  the  valve  in  the  steam  pipe  and  the  go- 
vernor. 

It  is  clear,  that  the  power  of  the  steam  engine  will  de- 
pend upon  the  energy  of  the  steam, — 1st.  Steam  of  two 
atmospheres  will,  other  things  being  equal,  produce  double 
the  effect  of  steam  of  one  atmosphere. — 2d.  the  force  of  the 
steam  remaining  the  same,  the  power  of  the  engine  will 
depend  on  the  extent  of  surface  acted  upon,  that  is,  on  the 
area  of  the  piston. — 3rf.  these  two  circumstances  remaining 
the  same,  the  power  of  the  engine  will  depend  on  the 
velocity  with  which  the  piston  moves. 

For  the  sake  of  illustration,  let  us  suppose  that  steam  is 
admitted  into  the  cylinder,  so  as  to  press  down  the  piston 
with  the  force  of  one  hundred  pounds,  and  that  the  length 
of  the  stroke  is  five  feet ;  and  suppose  that  the  end  of  the 
piston  red  is  attached  to  a  beam  whose  fulcrum  is  in  the 
Jcentre,  and  that  to  the  other  end  of  the  beam  there  is 
attached  a  weight  of  any  thing  less  than  one  hundred 


252  STEAM    ENGINE. 

pounds,  there  being  no  friction.  By  the  descent  of  the 
piston,  the  weight  at  the  end  of  the  beam  will  be  raised  5 
feet;  therefore  it  follows,  that  100  pounds  raised  5  feet 
during  one  descent  of  the  piston,  will  express  the  mechanic- 
al effect  of  the  engine.  The  reader  will  easily  perceive 
that  the  weight  at  the  end  of  the  beam  must  be  somewhat 
less  than  100  pounds,  for  as  it  acts  contrary  to  the  power 
of  the  piston,  if  they  were  equal  the  machine  would  be  at 
rest.  If  we  suppose  the  area  of  the  piston  double  of  what 
t  was  before,,  other  things  being  the  same,  the  engine 
would  raise  200  pounds  through  the  same  space  of  5  feet 
in  the  same  time :  and  the  same  effect  would  evidently 
ensue  if  we  supposed  the  area  of  the  piston  to  remain  as  it 
was  at  first,  but  the  force  of  the  steam  to  be  doubled.  If 
the  area  of  the  piston  and  force  of  steam  be  the  same  as  at 
first,  but  the  length  of  stroke  doubled,  then  the  mechanical 
effect  of  the  engine  will  be  100  Ibs.  raised  10  feet  high 
during  one  descent  of  the  piston ;  and  if  the  descents  be 
performed  in  the  same  time,  this  engine  will  be  double  the 
power  of  the  first. 

Let  us  proceed  now  to  actual  cases.  In  the  common 
low-pressure  steam  engine  of  Watt,  steam  is  admitted  into 
the  cylinder  whose  elastic  force  is  somewhere  about  that 
of  the.  atmosphere,  which  we  have  all  along  supposed  to  be 
15  Ibs.  to  the  square  ir.ch ;  but  friction  and  imperfect 
vacuums  tend  to  diminish  this  pressure,  and  the  effective 
pressure  may  be  reckoned  only  four-fifths  of  this.  If  the 
pressure  of  the  steam  is  diminished  by  its  one-fifth  part, 
which  is  3  Ibs.  to  the  square  inch,  then  will  the  effective 
pressure  be  12  Ibs.  to  the  square  inch.  The  working 
pressure  is  generally  reckoned  at  10  Ibs.  to  the  circular 
inch,  and  Smeaton  only  makes  it  7  Ibs.  The  effective 
pressure  we  have  taken  is  between  these  extremes,  being 
equivalent  to  9-42  Ibs.  to  the  circular  inch. 

Mr.  Tredgold  gives  the  following  table,  which  will  show 
how  the  power  of  the  steam,  as  it  issues  from  the  boiler,  is 
distributed.  In  an  engine  which  has  no  condenser : 

The  pressure  on  the  boiler  being lO'OOO 

1.  The  force   necessary  for  producing 
motion  of  the  steam  in  the  cylinder- •   *0069 

2.  By  cooling  in  the  cylinder  and  pipes  -0160 

3.  Friction  of  piston  and  waste '2000 


STKA.M    iJNGJ.Nj;.  253 

4.  The  force  required  to  expel  the  steam 

into  the  atmosphere '0069 

5.  The  force  expended  in  opening  the 
valves,  and  friction  of  the  parts  of  an 
engine -0622 

6.  By  the  steam  being  cut  off  before  the 

end  of  the  stroke -1000 

Amount  of  deductions  3920 

Effective  pressure- •••  6080 

In  one  which  has  a  condenser : — 
The  pressure  on  the  boiler  being 1000 

1.  By  the  force  required  to  produce  motion 

of  the  steam  into  the  cylinder 007 

2.  By  the  cooling  in  the  cylinder  and  pipes  016 

3.  By  the  friction  of  the  piston  and  loss  ••••   125 

4.  By  the  force  required  to  expel  the  steam  ; 
through  the  passages  007 

5.  By  the  force  required  to  open  and  close 
the  valves,  raise  the  injection  water,  and 
overcome  the  friction  of  the  axes 063 

6.  By  the  steam  being  cut  off  before  the  end 

of  the  stroke 100 

7.  By  the  power  required  to  work  the  air- 
pump 050 

368 

632 

If  we  now  suppose  a  cylinder  whose  diameter  is  24  inches, 
the  area  of  this  cylinder,  and  consequently  the  area  of  the 
piston  in  square  inches,  will  be, 

24a  x  '7854  =  452-39. 

Let  us  also  make  the  supposition  that  steam  is  admitted 
into  the  cylinder  of  such  power  as  exerts  an  effective  pres- 
sure on  the  piston  of  12  Ibs.  to  the  square  inch  ;  therefore, 
452-39  X  12  =  5428-68  Ibs.,  the  whole  force  with  which 
the  piston  is  pressed.  If  we  now  suppose  that  the  length 
'  of  the  stroke  is  five  feet,  and  the  engine  makes  44  single 
or  22  double  strokes  in  a  minute,  then  the  piston  will  move 
through  a  space  of  22  x  5  x  2  =  220  feet  in  a  minute : 
<end  from  what  has  been  said  before,  it  will  not  be  difficult 
to  see,  that  the  power  of  the  engine  will  be  equivalent  to  a 
-weight  of  5428  Ibs.  raised  through  220  feet  in  a  minute. 
22 


254  STEAM    ENGINE. 

This  is  the  most  certain  measure  of  the  power  of  a  steam 
.engine.  It  is  usual,  however,  to  estimate  the  effect  as  equi 
valent  to  the  power  of  so  many  horses.  This  method, 
however  simple  and  natural  it  may  appear,  is  yet,  from 
differences  of  opinion  as  to  the  power  of  a  horse,  not  very 
accurate  ;  and  its  employment  in  calculation  can  only  be 
accounted  for  on  the  ground,  that  when  steam  engines  were 
first  employed  to  drive  machinery,  they  were  substituted 
instead  of  horses  ;  and  it  became  thus  necessary  to  estimate 
what  size  of  a  steam  engine  would  give  a  power  equal  to 
so  many  horses. 

There  are  various  opinions  as  to  the  power  of  a  horse. 
According  to  Smeaton,  a  horse  will  raise  22,916  Ibs.  one 
foot  high  in  a  minute.  Desaguliers  makes  the  number 
27,500 ;  and  Watt  makes  it  larger  still,  that  is,  33,OOC 
There  is  reason  to  believe  that  even  this  number  is  too 
small,  and  that  we  may  add  at  least  11,000  to  it,  which  gives 
44,000  Ibs.  raised  one  foot  high  per  minute. 

Now,  in  the  case  above,  we  found  that  the  engine  of  24 
inch  cylinder,  would  raise  5428  Ibs.  through  the  space  of 
220  feet  in  one  minute  ;  and  it  is  easily  seen  that  it  could 
raise  220  x  5428  Ibs.  through  one  foot  in  the  same  time, 
therefore,  220  X  5428  =  1194160  Ibs.  raised  through  one 
foot  in  one  minute,  is  the  effective  power  of  the  engine ; 
and  from  these  considerations  it  will  be  easy  to  find  the 
power  according  to  the  different  estimates  of  a  horse's 
power.  For, 

1194160 
22Q16    =  52  horses   power, 

according  to  Smeaton. 
1194160 

TTSOO"  =  43  horses  P°wer' 

according  to  Desaguliers. 
1194160 

33000"  =  3 
according  to  Watt. 

H94160 
44000      =  27  horses' power, 

according  to  the  usual  estimate. 

The  reader  will  have  no  difficulty  in  forming  a  general 
rule  for  estimating  the  power  of  a  steam  engine.  (The 


STEAM    ENGINE.  255 

effective  pressure  on  each  square  inch  x  the  area  of  piston 
in  square  inches  X  length  of  stroke  in  feet  X  number  of 
strokes  per  minute)  -5-  44000  =  the  number  of  horses' 
ppwer  of  the  engine. 

What  is  the  power  of  a  low-pressure  engine,  whose 
cylinder  is  30  inches  diameter,  length  of  stroke  6  feet, 
making  16  double  strokes  in  the  minute  ? 

NOTE. — An  easy  rule  to  find  the  area  of  the  piston  in 
s>quare  inches,  is  this, 

The  diameter  x  circumference 

-  =  area. 
4 


Here  we  have, 

30  X  (30  x  3-1416)        2827-44 


=  706-86, 


4  4 

equa  the  area  of  the  piston  in  square  inches  ;  and  12  the 
effective  pressure,  6  the  length  of  stroke,  16  the  number  of 
double  strokes  in  a  minute  ? 

706  86  Xl2x6xl6x2_  1628605*44  _ 

44000~~  44000 

horses'  power. 

If  the  cylinder  of  a  higrh-pressure  steam  engine  has  a 
piston  of  5  inches  diameter,  with  a  twelve  inch  stroke, 
making  32  double  strokes  in  a  minute  ;  steam  being  ad- 
mitted of  an  elastic  force  equivalent  to  7  atmospheres  on  the 
inside  of  the  cylinder.  Its  effective  pressure  will  be  7  X 
15  =  105  Ibs.  to  the  square  inch  without  friction;  but  al- 
lowing one-fifth  for  friction,  the  effective  pressure  will  be 
105  —  21  =  84  Ihs.  to  the  square  inch. 

5  X  (3-1416  X  5) 
here  -  -  -     -  =  19-63  the  area  of  the  piston  : 

19-63  X  84  x  1  X  32  x  2  _  105530-88 

44000  "    44000 

horses',  power. 

A  convenient  rule  for  finding  the  power  of  a  high-pres- 
sure engine,  is  —  let  P  be  the  force  of  the  steam  in  the 
boiler,  A  the  area  of  the  piston,  and  V  the  velocity  of  the 
piston  in  feet  per  minute,  then, 
0-9  P  _  6  x  A  X  V 


44000 

^  The  pressure  of  the  steam  in  a  boiler  is  30  Ibs.  per 
square  inch,  the  diameter  of  cylinder  12  inches,  length  of 


25tl  STEAM    ENOINE. 

stroke  3  feet,  and  the  engine  making  30  double  strokes  put 
minute.  Here  the  area  of  piston  will  be  113-097,  tho 
velocity  of  piston  =  3  x  30  X  2  =  180  feet  per  minute 
and  since  0-9  x  30  —  6  =  21,  then, 

0-9  x  30  —  6  X  113-097  X  180        427506-66 


44000 
9*7  horses'  power. 

We  might  simplify  this  rule  still  farther  on  the  consi 
deration,  that  the  divisor  44000  may  be  viewed  as  the  de 
nominator  of  a  fraction  whose  numerator  is  one,  and  by 
converting  this  into  a  decimal,  and  multiplying  by  it,  we 
might  avoid  the  necessity  of  division. 

Since  —  —  -  =  -0000227,  hence  we  may  devise  the  rule. 
44000 

Effective  pressure  of  steam  X  area  of  piston  in  square 
inches  x  length  of  stroke  in  feet  x  number  of  strokes  per 
minute  x  227  ;  and  from  the  product  cutting  off  seven 
places  as  decimals  ;  —  the  horses'  power  of  the  engine. 

This  is  for  a  single  stroke  engine  —  for  a  double  stroke 
engine  the  multiplier  is  227  X  2  =  454. 

If  the  cylinder  be  42  inches  diameter,  and  the  piston 
moves  210  feet  per  minute,  then  the  engine  being  low 
pressure,  we  have, 

area  of  cylinder  equal  1385-44  ;  hence  227  X  1385-44  X 
210  X  12  =  792527097  : 

and  the  seven  figures  cut  off  as  decimals,  leave  79  horses' 
power. 

These  are  at  best  but  approximations,  and  for  safety  it 
might  be  advisable  that  a  lower-  number  than  12  should  be 
employed,  as  the  effective  pressure  of  the  steam  ;  the  num 
her  10  may  be  used  as  being  easily  managed,  and  coming 
near  the  truth  ;  and  thus  the  above  rule  may  be  simplified 
by  neglecting  the  pressure  of  the  steam,  and  cutting  off  six 
places  for  decimals  instead  of  seven,  as  there  is  reason  to 
believe  that  the  above  results  will  answer  only  ponies  in- 
stead of  strong  horses. 

The  stroke  of  an  engine  is  commonly  reckoned  equal  to 
one  complete  revolution  of  the  crank  shaft,  and  therefore 
double  the  length  of  the  cylinder,  and  it  has  been  stated 
by  Mr.  Thomas  Tredgold,  that  to  ascertain  the  velocity  of 
the  piston  when  the  engine  performs  at  its  maximum,  we 
may  employ  the  rule, 


STEAM    ENGINE.  257 

120  X  v/  length  of  stroke  =  velocity. 
If  an  engine  has  a  two  feet  stroke,  then, 

120  X  N/  2  =  120  x  1-4142  =  109-704, 
or  we  may  say  170,  as  the  velocity  of  the  piston  per  minute 
in  feet ;  wherefore  as  the  engine  has  a  single  stroke  of  2 
feet  we  have, 

170 

— —  =  42s  strokes  in  the  minute. 
4 

If  an  engine  have  a  four  feet  stroke,  then  we  have, 

120  X  v/  4  =  120  X  2  =  240  = 
the  velocity  of  the  piston  per  minute  ;  and, 

240 

•——  =  30,  equal  the  number  of  strokes  per  minute, 
o 

The  safety  valves  of  most  of  the  steam  engines  in  this 
part  of  the  country,  are  generally  loaded  with  a  weight  of 
from  3  to  4  Ibs.  to  the  square  inch  of  their  area;  let  us 
take  3  5  Ibs.  in  the  present  instance.  The  temperature  of 
steam  necessary  to  balance  this  pressure,  is,  according  to  the 
best  experiments,  223  degrees  of  Fahrenheit's  thermometer. 
But  besides  this  sensible  heat,  there  is  a  quantity  of  latent 
heat  not  indicated  by  the  thermometer,  and  which  can  only 
be  detected  when  the  steam  passes,  by  condensation,  into 
the  fluid  state  ;  as  the  latent  heat  is  then  given  out.  Now, 
if  the  latent  heat  of  the  steam  at  the  above  temperature,  be 
found  on  the  principle  stated  in  our  remarks  on  heat,  that  the 
sensible  and  latent  heats  of  steam  at  all  temperatures,  when 
added  together,  make  a  constant  quantity  ;  we  will  find  that 
the  latent  heat  of  steam  at  this  temperature  is  989.  The  real 
quantity  of  heat  then  in  the  steam  is  223  -f  989=  1212 
degrees.  We  will  not  be  far  from  the  truth  in  supposing, 
that  one  cubic  foot  of  this  steam  will,  when  condensed  into 
water,  measure  one  cubic  inch  ;  and  the  steam  is  supposed 
to  be  condensed  by  the  injection  of  cold  water.  Now  it  is 
evident,  that  the  temperature  of  the  water  formed  by  the 
condensation  of  the  steam,  will  be  somewhere  between  the 
temperature  of  cold  waierandthe  boiling  point.  Say  that 
the  temperature  of  the  injected  water  is  50  degrees,  and  that 
the  temperature  of  the  water  arising  from  the  condensation 
of  the  steam  is  100.  We  must  deduct  the  100  degrees  from 
Lj:e  heat  of  the  uncondensed  steam,  that  is,  1212 —  100  = 
H112,  which  is  left  to  be  communicated  to  the  injection 
22* 


258  STEAM    ENGINE. 

water  ,  and  since  each  cubic  inch  of  the  cold  water  requires 
50  of  heat  to  raise  it  to  the  temperature  of  the  water  found 
after  the  condensation  of  the  sieam,  therefore, 

1 1 12 

=  22T3,  cubic  inches 

50 

of  water  necessary  to  condense  one  cubic  foot  of  steam  to 
the  temperature  of  100,  the  injected  water  being  50. 

From  these  considerations  may  be  Serived  a  rule  for  de- 
termining the  quantity  of  water  necessary  to  condense  any 
quantity  of  steam,  at  any  given  temperature. 
Total  heat  of  the  steam  —  temperature  of  warm  water 

temp,  of  warm  water — temp,  of  cold  water 
quantity  of  steam  in  cubic  feet  =  the  quantity  of  cold  water 
in  cubic  inches  necessary  to  produce  the  effect. 

Let  us  illustrate  this  by  an  example. — What  quantity  of 
cold  water  will  it  require  of  the  temperature  of  60,  to  con- 
dense 8  cubic  feet  of  steam,  of  the  temperature  of  223,  to 
water  at  90  ?  The  whole  heat  is  as  before,  989  -f  223  = 
1212,  wherefore  by  the  rule, 

1212  —  90 

X  8  =  299-2  cubic  inches  = 


90 — 60 
299-2 


=  *17  of  a  cubic  foot  of  water. 


1728 

From  this  it  will  be  easy  to  determine  how  much  water 
must  be  discharged  by  the  pump  which  feeds  the  condenser, 
in  order  that  a  proper  vacuum  may  be  formed. 

From  practice  it  would  appear  that  about  26  cubic  inches 
of  cold  water  for  condensing  should  be  used  for  each  cubic 
foot  of  the  capacity  of  the  cylinder. 

We  may  infer  from  observation,  that  the  engines  com- 
monly in  use  require  betwixt  3|  and  4  gallons  of  cold  water 
per  minute  for  each  horse's  power.  If  the  water  is  return- 
ed as  it  is  in  some  engines,  then  a  greater  quantity  will  be 
necessary.  Now,  in  the  usual  construction  of  engines,  the 
pump  rod  which  supplies  the  condenser  with  cold  water, 
is  fixed  halfway  between  the  end  of  the  beam  and  the  cen- 
tre ;  hence,  the  length  of  its  stroke  is  one-half  that  of  the 
piston  in  the  large  cylinder:  therefore,  if  there  be  a  40 
horse  power  engine,  the  length  of  whose  stroke  is  6  feet, 
the  length  of  the  stroke  of  the  pump  will  be  3  feet. 

Now  an  imperial  wine  gallon  occupies  a  space  of  277'274 


STKAM   KNGI.M:-  259 

cubic  inches,  and  71  Dillons  will  occupy  a  space  of  277'274 
X  7'5  =  2079-555  cubic  inches;  and  as  the  engine  is  40 
horses'  power,  there  must  be  discharged  in  one  minute, 

2079-555  x  40  =  83182-2  cubic  inches, 
and  if  the  engine  makes  30  strokes  per  minute,  then 

83182-2 

=  277-274  cubic  inches 

30 

discharged  at  one  stroke  :  but  the  stroke  is  3  feet  long,  and 
it  remains  only  to  find  what  must  be  the  diameter  of  a 
pump's  bore,  whose  length  is  36  inches,  so  that  its  capacity 
shall  be  2772  ;  hence  we  find  that, 

2772 

-^-  =  7/  inches, 

nearly  equal  to  the  area  of  the  pump's  bore  ;  now  the  area 
of  circles  are  to  each  other  as  the  squares  of  their  diame- 
ters, and  the  area  of  a  circle  whose  diameter  is  9,  is  63-6: 
therefore, 

63-6  :  77  : :  0-  :  JM, 

the  square  root  of  which  will  be  the  diameter  of  the  pump, 
and  will  be  found  =  9-9  inches. 

With  respect  to  the  fly  wheel, 

Horses'  power  of  engine  x  2000 

Velocity  of  circumfer.  wheel  in  feet  per  second- 
the  weight  of  the  fly  wheel  in  cwts. 

If  the  diameter  of  the  fly  of  a  30  horse  power  engine  be 
20  feet,  and  make  18  revolutions  per  minute,  then, 

20  X  3-1416  =  62-832  = 

circumference  in  feet,  and  62-832  X  18  =  1128-97  feet, 
the  space  which  the  circumference  moves  through  in  one 
minute ;  hence, 

1128-97 

— — =18-81  feet  per  second; 

30  x  2000       60000 

18-81-       =  353*  ==  169  CWtS' 
r=  8  tons  9  cwts.  the  weight  of  the  fly. 

In  the  working  of  the  valve  of  a  steam  engine,  an  eccentric 
wheel  is  often  employed,  and  it  becomes  necessary  to  cal- 
culate the  degree  of  eccentricity  necessary  to  give  a  certain 
length  of  stroke.  The  eccentric  wheel's  radius  mav  bo 


260  STEAM    ENGINE. 

easily  found;  thus,  suppose  the  length  of  stroke  required  is 
20  inches,  and  the  diameter  of  the  shaft  on  which  the  wheel 
is  screwed  is  5  inches,  and  the  thickness  of  metal  required 
to  key  on  the  wheel  21  inches.  Take  the  half  of  the  re- 
quired stroke,  that  is,  10  inches,  as  the  distance  of  the  cen- 
tre of  the  shaft  from  the  centre  of  required  wjieel,  and  add- 
ing to  this  the  half  thickness  of  the  shaft  =  2,1  inches,  as 
likewise  the  thickness  of  metal  necessary  for  keying  =  21, 
then  10  +  21  +  21  =  15  inches,  ihe  radius  of  the  wheel. 
Now  let  E  be  =  the  radius  of  the  eccentric  wheel  L  =  the 
length  of  the  eccentric  rod,  and  /  -=  the  length  of  the  bar 
between  the  other  end  of  this  rod  and  the  slide ;  and  let  e 
=  the  length  of  slide;  then, 

E  =  L_*_?  r   ==  /xE 

/  e 

I  x  E                             L  x  e 
P  — /  — 

L  E 

Suppose  the  length  of  the  stroke  of  the  slide  e  —  6  inches, 
the  length  of  the  slide  rod  /  =  5  inches,  and  the  radius  of 
the  eccentric  =  24  inches  =  E,  then  the  length  of  the  rod 

5  X  24 

L  =  =  20  inches. 

6  , 

The  other  rules  are  wrought  on  the  same  principle 
We  have  before  spoken  of  the  governor  while  treating  of 
central  forces  and  rotation.  It  remains  for  us  here  only 
to  observe,  that  the  governor  performs  in  one  minute  half 
as  many  revolutions  as  a  pendulum,  whose  length  is  the 
perpendicular  distance  between  the  plane  in  which  the  balls 
move  and  the  centre  of  suspension.  Thus,  if  the  distance 
between  the  point  of  suspension  and  the  plane  in  which  the 
balls  move  be  28  inches  : 

[/  39-1386  \ 

.]( — ^ )  =   I1 182  vibrations   in  a  second  from  the 

nature  of  the  pendulum  ;  hence, 

1-182 

— - —  =  0-591,  the  revolutions  of  the  governor  in  a  se- 
cond, or  0-591  x  60  =  35-46  in  one  minute. 

The  piston  rod  of  a  steam  engine  may  be  made  to  move 
up  and  down  in  a  right  line  in  various  ways.  The  rod  may 
be  made  to  terminate  in  a  rack,  the  teeth  of  which  act  in 


S1EAM    ENGINE  2611 

the  teeth  of  an  arched  head  of  the  .oAg  lever,  called  the 
working  beam  :  but  the  most  efficacious  of  all  contrivances 
of  this  kind,  is  that  of  Watt,  commonly  called  the  parallel 
motion  This  contrivance  is  founded  on  geometrical  prin- 
ciples, which  it  would  be  inconsistent  with  the  plan  of  this 
work  to  consider ;  we  shall  therefore  simply  describe  the 
contrivance  of  this  illustrious  mechanic. 

The  working  beam   has  an  alternating  circular  motion 
ound  its  centre  A,  and  it  is  clear  that  the  points  B  ami  G 

ill  have  a  circular  motion  round  the  common  centre  A. 

et  the  point  B  be  exactly  in  the  middle,  between  the 
centre  and  end  of  the  beam.  Let  there  be  a  bar  or  rod 
CD,  of  the  same  length  as  AB,  capable  of  moving  round 
the  centre  C,  by  means  of  a  pivot.  The  other  end  of  this 
rod  is"  attached  by  means  of  a  pivot,  to  the  rod  DB.  Now, 
by  the  alternate  rising  and  falling  of  the  beam,  the  points 


B  and  D  will  move  in  circular  arches,  but  the  middle  point 
P  of  the  connecting  rod  BD,  will  move  upwards  and 
djwnwards  in  a  vertical  straight  line,  or  at  least  so  very 
neatly  so,  as  the  difference  cannot  be  perceived.  Now,  to 
this  point  P,  there  is  attached  the  end  of  the  pump  rod, 
which  will,  of  course,  follow  the  direction  of  the  impelling 
point,  and  move  in  a  straight  line.  For  the  purpose  of 
communicating  a  similar  motion  to  the  other  piston  rod, 
conceive  another  rod  CP'  introduced,  of  the  same  length  as 
BD,  and  its  extremities  moving  likewise  on  pivots.  The 
oiston  rod  of  the  cylinder  is  attached  to  the  point  P',  and 
.this  point  moves  quite  in  the  same  way  as  the  point  P. 
The  only  difference  in  the  motion  of  these  two  points  will 
jje,  that  the  point  P'  will  move  twice  as  fast  as  the  point  P, 
or  will,  in  the  same  time,  move  twice  as  far. 


262  STEAM    ENGINE. 

The  length  of  the  links  are  made  =  4  to  5,  the  length  of 
the  stroke  being  1,  according  to  circumstances,  the  longef 
link  being  preferred  when  practicable.  From  the  length 
of  the  links  must  be  determined  the  position  of  the  radius 
bar,  for  the  vertical  distance  between  the  centres  of  motion 
of  the  working  beam  and  the  radius  bar  must  be  equal  to 
the  length  of  a  link. 

When  the  parallel  bar  is  not  more  than  one-half  of  the 
working  beam's  radius,  then, 

Let  B  =  radius  of  the  beam, 
P  =  length  of  parallel  bar, 
S  =  length  of  stroke, 
R  =  length  of  radius  bar;  we  have 

B  —  2Px(iS)"  R 

B  —  ^B'  —  [iS]2)  X2P  "* 

Suppose  the  length  of  the  beam  from  the  centre  =  12 
feet,  the  length  of  stroke  6,  and  of  parallel  bar  5  feet, 
that  is,  B  =  12,  S  =  6,  and  P  =  5,  then, 

B  —  2  P  X  (£S)a  =  12  —  10  X  (^6)3  =  18 
=  the  dividend  ;  then,  B  —  v/(B3  —  [|S]a)  x  2  P  =  12 
—  v/(12a  —  [16]2)    x   10  =  12  —  11-62   X   10  =  0-38 
X  10  =  3-8  the  divisor,  wherefore, 

18 

-   +  5  =  9-74  =  the  length  in  feet  of  the  radius  bar. 
3'8 

When  the  parallel  bar  is  more  than  half  the  length  of  the 
radius  of  the  beam,  the  rule  is, 

2P-BX(|S)» 
B  — (Ba  —  [|S]2)    x  2P 

by  which  rule  it  will  be  found  that  when  the  length  of 
stroke  and  radius  bar  are  each  6,  and  the  radius  of  beam  10 
feet,  the  length  of  radius  bar  will  be  2-75  feet. 

Many  rules  have  been  given  for  the  quantity  of  fuel 
necessary  for  the  production  of  steam,  but  they  cannot  be 
depended  on,  so  many  circumstances  must  be  taken  undei 
consideration — the  quality  of  material  used  for  fuel  and 
the  mode  of  constructing  the  fireplace. 

It  has  been  found  that  3  cwt.  of  Newcastle  coals  are 
equivalent  to  4  cwt.  of  Glasgow  co?ls,  or  9  cwt.  of  wood, 
or  7  cwt.  of  culm.  A  chaldron  of  coals  in  London  contains 
36  bushels,  and  weighs  3136  Ibs.,  or  nearly  1  ton.  8  cwt. 


. 

STEAM    ENGINE.  263 

It  would  appear,  that  in  the  common  low-pressure  steam 
engines,  the  eonsumpt  of  coal  per  hour  for  1  horse  power, 
is  about  16  Ibs.,  of  wood  56  Ibs.,-  and  of  culm  35  Ibs. 
These  statements  are  given  somewhat  large,  and  by  proper 
regulation  much  less  fuel  might  serve. 

In  the  boiler  there  are  certain  proportions  generally  ob- 
served. The  width,  depth,  and  length,  are  as  the  numbers 
1,  1-1,  2-5.  So  that  if  the  width  be  5  feet,  then  the  depth 
will  be  1-1  x  5  =  5  feet  6  inches  ;  and  the  length  5  x  2*5  = 
12  ft.  6  in. ;  and  the  whole  content  of  the  boiler  will  be, 

5  x  5-5  X  12-5  =  343-75  cubic  feet. 
Now  Boulton  and  Watt  allow  25  cubic  feet  of  space  in  the 
boiler  for  each  horse  power  ;  and  according  to  this  estimate, 

343'75 

— — —  =  13  and  a  fraction,  the  number  of  horses'  power 
•0 

of  this  engine  for  which  this  boiler  would  be  fitted.  Some, 
instead  of  computing  the  size  of  boiler  in  this  way,  allow 
5  square  feet  of  surface  of  water  for  each  horse's  power; 
but  in  all  cases,  it  is  common  to  make  the  boiler  of  a  size 
fitted  for  an  engine  of  at  least  2  horses'  power  more  than 
that  to  which  it  is  applied. 

There  are  two  ways  of  loading  the  safety  valve  of  a 
boiler  ;  the  one  by  placing  a  weight  on  the  top  of  it,  and 
the  other  by  causing  the  weight  to  act  on  the  valve  by  a 
lever. 

When  the  weight  is  placed  upon  the  valve ;  area  of 
valve  X  pressure  per  square  inch  =  whole  weight,  and  also 

whole  weight 

—  =  pressure  per  square  inch. 

area  of  valve 

Thus,  if  a  weight  of  50  Ibs.  be  placed  upon  a  valve  whose 
area  is  10  inches,  then  the  pressure  per  square  inch  is 

—  =  5  Ibs.  pressure  per  square  inch. 

When  the  weight  acts  by  a  lever,  it  is  placed  at  one  end 
the  fulcrum  being  at  the  other,  and  the  valve  connected 
with  the  lever  somewhere  between  them ;  this,  then,  is  a 
simple  case  of  the  lever.  Hence,  if  the  length  of  the  lever 
be  24  inches,  the  diameter  of  the  valve  3  inches,  (its  area 
will  be  7,)  the  distance  between  the  fulcrum  and  the  valve 
3  inches,  then  to  give  60  Ibs.  pressure  per  square  inch  on 
Ahe  valve  60  x  7  =  420  Ibs.  the  whole  pressure  on  the 
valve,  and 


264 


STEAM    ENGINE, 


420  x  3 


60  Ibs.  will  be  the  weight  hung  at  the 


24  —  3 

end  of  the  lever  to  give  the  required  pressure. 

To  find  the  action  of  the  weight  of  the  lever  divide  its 
whole  length  by  the  distance  of  the  valve  from  the  fulcrum, 
and  multiply  the  quotient  by  half  the  weight  of  the  lever. 

The  following  rules  for  calculations  connected  with  the 
steam  engine  are  extracted  from  a  useful  little  compendium 
lately  published  by  Mr.  Templeton,  of  Liverpool.  These 
rules  we  have  inserted  here,  not  so  much  for  their  superior 
accuracy,  as  from  a  desire  to  present  our  readers  with 
methods  by  which  they  may  approximate  to  the  true  results 
by  means  of  the  sliding  rule.  It  is  to  be  observed  that  the 
term  gauge  point  is  used  to  denote  the  number  to  be  taken 
on  the  line  stated  in  the  rule. 


Length  of 
strtfke   in 
ft.  and  in. 

Gauge  point. 

Length  of 
stroke  in 
ft.  and  in. 

Gauge  point. 

2     0 

295 

6     0 

392 

2     6 

318 

7     0 

41 

2     9 

322 

8     0 

414 

3     0 

33 

MARINE    ENGINES. 

3     6 

335 

3     0 

3 

4     0 

343 

3     6 

31 

4     6 

355 

4     0 

317 

5     0 

385 

4     6 

326 

RULE. — Set  the  gauge  point  upon  C  to  1  upon  D,  and 
against  the  number  of  horses'  power  upon  C,  is  the  diame- 
ter in  inches  upon  D  ;  or,  against  the  diameter  in  inches 
upon  D,  is  the  number  of  horses'  power  upon  C. 

Ex.  1. — What  diameter  must  a  cylinder  be  with  a  4  feet 
stroke,  to  be  equal  to  20  horses'  power? 

Set  343  upon  C  to  1  upon  D  ;  and  against  20  upon  C  is 
24'2  inches  diameter  upon  D.. 

Ex.  2. — What  number  of  horses'  power  will  an  engine 
be  equal  to,  when  the  cylinder's  diameter  is  19  inches  and 
stroke  3  feet  ? 

193  X  "7854  x  7-25  x  192  _  394672-7328  _ 

33000  33000 

11*96  or  12  horses'  power  nearly. 


i. 


STEAM    ENGINE.  265 

The  proportion  of  parts  of  a  high-pressure  steam  en- 
gine.— The  U:ngih  of  the  stroke  should,  if  possible,  be 
twice  its  diameter.  'The  velocity  in  feet  per  minute  should 
be  103  times  the  square  root  of  the  length  of  the  stroke  in 
feet.  Anil,  as  4S-M)  is  to  the  velocity  thus  found,  so  is  the 
area  of  the  cylinder  to  the  area  of  the  steam  passages. 

The  proportions  of  the  parts  of  an  atmospheric  en- 
gine.— The  length  of  the  cylinder  should  be  twice  the  dia- 
meter. The  velocity  in  feet  per  minute  should  be  ninety- 
eight  times,  the  square  root  of  the  length  of 'the  stroke  in 
feet.  The  area  of  the  steam  passages  will  be  as  4800  is  to 
the  velocity  in  feet  per  minute,  so  is  the  area  of  the  cylin- 
der to  the  area  of  the  steam  passage.  If  the  area  of  the 
"  cylinder  in  feet  be  multiplied  by  half  the  velocity  in  feet, 
and  that  product  by  1"23  added  to  1'4  divided  by  the  dia- 
meter in  feet,  the  result  divided  by  1480  will  give  the  cubic 
feet  of  water  required  for  steam  per  minute.  If  the  num- 
ber of  times  the  quantity  of  water  required  for  injection 
must  be  greater  than  that  required  for  steam,  in  general  it 
will  be  about  twelve  times  the  quantity,  but  it  had  better  be 
a  little  in  defect  than  excess.  The  aperture  for  the  injec- 
tion must  be  such  that  the  above  quantity  of  water  will  be 
injected  during  the  time  of  the  stroke.  In  order  that  the 
injection  be  sufficiently  powerful  at  first,  the  head  should 
be  about  three  times  the  height  of  the  cylinder  ;  and  making 
the  jet  apertures  square,  the  area  should  be  the  850th  part 
of  the  area  of  the  cylinder.  The  conducting  pipe  should 
be  about  four  times  the  diameter  of  the  jet. 

The  piuportions  of  the  parts  of  a  single-acting  low- 
pressure  engine. — The  length  of  the  cylinder  should  be 
twice  its  diameter.  The  velocity  of  the  piston  in  feet  per 
minute  should  be  ninety-eight  times  the  square  root  of  the 
length  of  the  stroke.  The  area  of  the  steam  passages  should 
be  equal* to  the  area  of  the  cylinder,  multiplied  by  the  velo- 
city of  the  piston  in  feet  per  minute,  and  divided  by  4800. 
The  air  pump  should  he  one-eighth  of  the  capacity  of  the 
cylinder,  or  half  the  diameter  and  half  the  length  of  the 
strike  of  the  cylinder,  and  the  condenser  should  be  of 
the  same  capacity.  The  quantity  of  steam  will  be  found 
bv  multiplying  the  area  of  the  cylinder  in  fe«t  by  half  the 
velocity  in  feet;  with  an  addition  of  one-tenth  for  cooling 
f  ami  waste,  and  this  divided  by  the  volume  of  the  steam 
corresponding  to  its  force  in  the  boiler,  gives  the  quantity 
23 


266  RAILWAYS. 

of  water  required  for  steam  per  minute,  from  whence  the 
proportions  of  the  boiler  may  be  determined.*  At  the  com- 
mon pressure  of  two  pounds  per  circular  inch  on  the  valve, 
the  divisor  will  be  1497.  The  quantity  of  injection  water 
should  be  twenty-four  times  that  required  for  steam,  and 
the  diameter  of  the  injection  pipe  one-thirty-sixth  of  the 
diameter  of  the  cylinder.  The  valves  in  the  air  pump  bucket 
should  be  as  large  as  they  can  be  made,  and  the  discharge 
and  foot  valves  not  less  than  the  same  area. 

Summary  of  proportions  of  a  double  engine,  working 
at  full  pressure. — The  length  of  a  cylinder  should  be  twice 
its  diameter ;  for  a  cylinder  having  this  proportion  exposes 
less  surface  to  condensation  than  any  other  enclosing  the 
same  quantity  of  steam.  The  area  of  the  steam  passages  * 
should  be  about  one-fifth  of  the  diameter  of  the  cylinder; 
or  their  area  should  be  equal  to  the  area  of  the  cylinder, 
multiplied  by  the  velocity  of  the  piston  in  feet  per  minute, 
and  divided  by  4800.  The  diameter  of  the  air  pump  should 
be  about  two-thirds  of  the  diameter  of  the  cylinder,  and 
half  the  length  of  stroke ;  and  the  larger  the  passages 
through  the  air  bucket  and  the  discharging  flap  are,  the 
better.  The  quantity  of  water  for  injection  should  be  about 
283  times  that  required  for  steam,  or  about  26  cubic  inches 
to  each  cubic  foot  of  the  contents  of  the  stroke  of  the  pis- 
tog.  Watt  considered  a  wine  pint,  or  28g  cubic  inches, 
quite  sufficient.  There  should  be  62  times  as  much  water 
in  the  boiler  as  is  introduced  at  one  feed. 

These  proportions  are  taken  from  Tredgold's  valuable 
treatise  on  the  steam  engine. 


RAILWAYS,  STEAMBOATS,  &c. 

IT  has  been  deduced  from  very  extensive  experiments  or 
the  Liverpool  and  Manchester  railways,  that  the 'effective 
power  of  a  locomotive  engine  is  about  '3  of  the  pressure  of 
the  steam  on  the  piston,  on  the  calculated  power  of  the 
engine  being  1.  In  one  case,  for  instance,  a  cylinder  21 
inches  diameter  was  used,  the  elasticity  of  steam  in  the 
boiler  was  30  Ibs.  to  the  square  inch,  above  the  pressure 

*  To  459  add  the  temperature  in  degrees,  and  multiply  the  sum  by 
76-5.  Divide  the  product  by  the  force  of  the  steam  in  inches  of  mer- 
cury, and  the  result  will  be  the  space  in  feet  the  steam  of  a  cub'c  foot 
of  water  will  occupy. 


RAILWAYS.  .  -867 

ol  the  atmosphere.  The  length  of  the  rail,  which  was  in- 
clined, \vas  :H65  feet,  and  the  height  24  feet.  The  time 
of  drawing  6  loaded  wagons,  each  weighing  9010  Ibs.  up  the 
rail,  was  570  seconds,  during  which  time  the  engine  made 
444  single  strokes,  each  5  feet  long.  Now, 

21s  x  '7854  =  346-36  =  the  area  of  the  piston  in  square 
inches,  wherefore,  346'36  x  30  =  10390  Ihs.  =  the  pres- 
sure of  steam  upon  the  piston,  whose  stroke  was  5  feet, 
and  number  of  strokes  in  the  given  time  444;  hence  444  x 
3  ==  2220  feet  =  the  space  through  which  the  power  10390 
has  traversed;  therefore,  10390  x  2220  =  23065800  Ibs. 
=  the  impelling  power  of  the  engine.  Now,  it  was  found 
that  the  actual  weight  including  resistance  moved,  wa* 
'7124415  Ihs.;  then, 

7124415 

which  will  give  the  effect  about  30-9  per 
230uobOO 

cent.,  but  the  foregoing  number  may  be  taken  as  a  safe  me- 
dium, that  is,  30  per  cent  or  -3. 

The  amount  of  retardation,  arising  from  steam  carriages 
moving  on  railways,  has  been  estimated  thus ; 

Loaded  carriages  weighing  altogether  8522  Ibs.  the  fric- 
tion amounted  to  50  Ibs.,  or  the  ^-$  part  of  the  weight.  In 
empty  carriages  weighing  2586  Ibs.,  the  friction  amounted 
to  10  Ibs.,  or  the  ^TS  Part  °f  tne  weight;  and  the  friction 
may  be  regarded  as  a  constant  retarding  force.  Wrought 
iron  rails  seem  from  a  multitude  of  experiments  to  be  much 
better  than  those  of  cast-iron,  as  they  are  more  durable  and 
cause  less  friction. 

The  Rocket  was  tried,  weighing  4  tons  and  5  cwt.,  to  it 
there  was  attached  a  tender  with  water  and  coals,  weighing 
3  tons,  2  cwt.  0  quar.  2  Ibs. ;  and  two  carriages  loaded 
with  stones,  weighing  9  tons,  10  cwt.  3  qr.  26  Ibs.,  making 
in  all  .17  tons.  At  full  speed  she  moved  at  the  rate  of  30 
miles,  in  2  hours,  6  minutes,  9  seconds,  or  14|  per  hour 
at  the  end  of  stage,  about  6  miles ;  and  the  greatest  velocity 
was  29 1  miles  per  hour.  The  quantity  of  water  used  92'6 
cubic  feet,  and  it  required  11/j  Ibs.  of  coke  for  each  cubic 
foot  of  steam. 

In  the  Rocket  the  boiler  is  cylindrical,  with  flat  ends  6 
feet  long,  and  3  feet  4  inches  in  diameter.  To  one  end  of 
the  boiler  there  is  attached  a  square  box  as  a  furnace,  3  feet 
Jong  by  2  feet  broad,  and  about  3  feet  deep — at  the  bottom 
of  this  box  five  bars  are  placed,  and  the  box  is  entirely 
surrounded  with  a  casting,  except  at  the  bottom  and  the 


"8  RAILWAYS. 

side  next  the  boiler.  Betwixt  the  casting  and  the  box 
there  is  left  a  space  of  about  3  inches,  which  is  kept  con- 
stantly filled  with  water.  The  upper  half  of  the  boiler  is 
used  .as  a  reservoir  for  steam;  the  under  half  being  kept 
filled  with  water,  and  through  this  part  copper  tubes  reach 
from  one  end  to  the  other  of  the  boiler,  being  open  to  the 
fire  box  at  one  end,  to  the  chimney  at  the  other ;  these 
tubes  are  25  in  number,  each  being  3  inches  in  diameter. 
The  cylinders  were  each  8  inches  in  diameter,  and  one  was 
at  each  side  of  the  boiler;  the  piston  had  a  stroke  of  16^ 
inches.  The  diameter  of  the  large  wheels  was  4  feet  85 
inches.  The  area  of  the  surface  of  water,  exposed  to  the 
radiant  heat  of  the  fire,  was  20  square  feet,  being  that  sur 
rounding  the  fire  box  or  furnace ;  and  the  surface  exposed 
to  the  heated  air  or  flame  from  the  furnace,  or  what  may 
be  called  communicative  heat,  is  117'8  square  feet. 

The  average  velocity  of  the  Rocket  may  be  stated  at  14 
miles  per  hour,  and  during  one  hour  she  evaporates  18'24 
cubic  feet  of  steam,  with  a  consumpt  of  about  17'7  Ibs.  of 
coke  for  each  cubic  foot  of  water. 

An  empirical  rule  has  been  given  for  the  ascertaining  of 
the  quantity  of  fuel  necessary  for  steam  carriages,  which 
may  be  useful. 

The  weight  of  the  load  X  51 '55  -f  weight  of  carriages 

898 

the  quantity  of  coals  required  to  carry  one  mile,— but  a  near 
approximation  to  the  truth  may  be  to  allow  2  Ibs.  for  every 
ton  for  one  mile. 

Iron  railroads  are  of  two  descriptions.  The  flat  rail,  or 
tram  road,  consists  of  cast  iron  plates  about  3  feet  long, 
4  inches  broad,  and  5  an  inch  or  1  inch  thick,  with  a 
flaunch,  or  turned  up  edge,  on  the  inside,  to  guide  the 
wheels  of  the  carriage.  The  plates  rest  at  each  «nd  on 
Ptone  sleepers  of  3  or  4  cwt.  sunk  into  the  earth,  and  they 
are  joined  to  each  other  so  as  to  form  a  continuous  horizon- 
tal pathway.  They  are,  of  course,  double;  and  the  distance 
between  the  opposite  rails  is  from  3  to  4|  feet,  according 
to  the  breadth  of  the  carriage  or  wagon  to  be  employed 
The  edge  rail,  which  is  found  to  be  superior  to  the  tram 
rail,  is  made  either  of  wrought  or  cast  iron  ;  if  the  latter  b*1 
used,  the  rails  are  about  3  feet  long,  3  or  4  inches  broad, 
and  from  1  to  2  inches  thick,  being  joined  at  the  ends  by 
cast  metal  sockets  attached  to  the  sleepers.  The  upper 
edge  of  the  rail  is  generally  made  with  a  convex  surface 


RAILWAYS. 


269 


to  which  the  wheel  of  the  carriage  is  attached  by  a  groove 
made  somewhat  wider.  When  wrought  iron  is  used,  which 
is  in  many  respects  preferable,  the  bars  are  made  of  a 
smaller  size,  of  a  wedge  shape,  and  from  12  to  18  feet 
long;  but  they  are  supported  by  sleepers,  at  the  distance 
of  every  3  feet.  In  the  Liverpool  railroad  the  bars  are  15 
feet  long,  and  weigh  35  Ibs.  per  lineal  yard.  The  wagons 
in  common  use  run  upon  4  wheels  of  from  2  to  3  feet  in 
diameter.  Railroads  are  either  made  double,  1  for  going 
and  1  for  returning;  or  they  are  made  with  sidings^  where 
the  carriages  may  pass  each  other. — See  M'Cullocli's  Diet. 

Table  showing  the  effects  nf  a  force  of  traction  of  100  pounds,  ai 
different  velocities,  on  canals,  railroads,  and  turnpike  roads.* 


Velocity  of  motion. 

Load  moved  by  a  power  of  100  ll». 

Miln  per 
hour. 

Feet  per 

second. 

On  a  Canal. 

On  a  level  Railway. 

On  a  level 
Turnpike  Road. 

Total  mass 
moved. 

UKful  ef- 
fect. 

Total  man 
moveJ  . 

L'Kfal  ef- 
fect. 

Total  man  !  Dseful  ef- 
moved.     I       feet. 

2£ 
3 

3-66  55,500  39,400 
4-40  38,542  27,361 

Ibt. 

14,400 
14,400 

10,800 
10,800 

Us.       i        Hi. 

1,800.1,350 
1,800  1,350 

3d 

5-13  28,316  20,100 

14,400 

10,800 

1,800  1,350 

4 

5-86  21,680  15,390 

14,400  10,800 

1,800  1,350 

5 

7-33  13,875     9,850 

14,400  10,800 

1,800  1,350 

6 

8-80    9,635 

6,840 

14,400  10,800 

1,800  1,350 

7 

10-26    7,080 

5,026 

14,400  10,800 

1,800  1,350 

8 

11-73 

5,420 

3,848 

14,400  10,800 

1,800  1,350 

9 

13-20 

4,282     3,040 

14,400  10,800 

1,800  1,350 

10 

14-66!  3,468 

2.462 

14,400  10,800  1,800  1,350 

13-5 

19-9 

1,900 

1,359 

14,400 

10,800  1,800,1,350 

The  subject  of  steam  vessels  has  been  investigated  by 
different  engineers,  on  mathematical  principles,  but  the 
calculations  which  their  rules  direct  are  by  far  too  intricate 
for  a  work  of  this  nature.  We  will,  however,  insert  a  state- 
ment of  the  proportions,  &c..  of  several  steamboats  already 
made,  which  will  doubtless  be  acceptable  to  the  practical 
man,  and  those  who  wish  to  investigate  the  theory  will  find 
'  ample  material  in  the  work  of  Tredgold. 

*  The  force  of  traction  on  a  canal  varies  a<  the  square  of  the  velocity 
but  the  mechanical  power  necessary  to  move  the  boat  is  usually  reckoned 
Ao  increase  as  the  cube  of  the  velocity.      On  a  railroad  or  turnpike,  the 
force  of  traction  is  constant;   but  the  mechanical  power  necessary  to 
move  the  carriage,  increases  as  the  velocity. 

23* 


870 


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ANIMAL    STRENGTH.  273 

The  rule  for  determining  the  tonnage  is  according  to  law. 
but  by  no  means  according  to  correct  principles.  It  is  as 
follows  :  — 

Take  the  length  =  L  from  the  back  of  the  main  stern 
?ist  to  the  fore  part  of  the  main  stem<  beneath  the  bow- 
sprit, and  subtract  from  it  the  length  of  the  engine  room 
=  E,  and  from  the  remainder  subtract  three-fifths  of  B  = 
the  breadth  of  the  vessel  taken  from  outside  to  outside  of 
the  planks  at  the  widest  part  of  the  vessel,  whether  it  be 
above  or  below  the  wales,  and  divide  this  last  remainder  by 
188  ;  the  quotient  multiplied  by  the  square  of  B  will  give 
the  register  tonnage,  or, 


Wherefore  the  length  being  162  feet,  the  length  of  engine 
-oom  47,  and  the  breadth  of  the  vessel  32,  then, 


tonnage. 


ANIMAL   STRENGTH. 

% 

THERE  is  a  certain  load  which  an  animal  can  just  bear 
but  cannot  move  with  it,  and  there  is  a  certain  velocity  with 
which  an  animal  can  move  but  cannot  carry  any  load.  In 
these  two  circumstances  it  is  clear,  that  the  exertion  of  the 
animal  can  be  of  no  avail  as  a  mover  of  machinery.  These 
are,  as  it  were,  the  extremes  of  the  animal's  exertion, 
where  its  effect  is  nothing  ;  but  between  these  two  extremes, 
there  must  be  weights  and  velocities  with  which  the  animal 
can  move,  and  be  more  or  less  efficient. 

If  one  man  travel  at  the  rate  of  three  miles  an  hour,  and 
jarry  a  load  of  56  Ibs.,  and  another  move  at  the  rate  of  4 
miles  an  hour  and  carry  a  load  of  42  Ibs.,  the  speed  of 
the  first  is  3,  and  the  load  56,  the  useful  effect  may  there- 
fore be  estimated  as  the  momentum  =  168.  The  other 
Carries  only  42  Ibs.,  but  travels  at  the  rate  of  4  miles  an 
hyur ;  therefore,  in  the  same  way,  his  useful  effect  will  be 


nyur ;  tnei 
r  x  42  = 


X  42  =  168,  the  same  as  before:  hence  the  effect  of 


274  ANIMAL    STRENGTH. 

these  twc  men  are  the  same.  It  will  not  be  difficult  to 
show,  that  in  the  same  time  they  perform  the  same  quantity 
of  work.  For  the  first  will  in  six  hours  carry  56  Ibs.  3x6 
=  18  miles,  as  he  travels  at  the  rate  of  3  miles  an  hour; 
and  if  he  be  supposed  to  carry  a  different  load,  hut  of  the 
3ame  weight  every  rnile,  he  will  in  the  six  hours  have  car- 
ried altogether  18  X  56  =  1008  Ibs.  ;  but  the  other  carries 
in  the  same  way,  4  times  42  Ibs.  every  hour,  that  is  168 
Ibs.  in  one  hour — therefore  in  6  hours  he  will  have  carried 
168  x  6  =  1008  Ibs.,  the  same  as  the  other. 

It  will  now  be  seen  what  is  meant  by  the  phrase  useful 
effect,  and  from  what  has  been  observed  above,  we  will  be 
led  to  conclude,  that  when  the  load  is  the  greatest  which 
the  animal  can  possibly  bear ;  the  useful  effect  is  nothing, 
because  the  animal  cannot  move  ;  and  when  the  animal 
moves  with  its  greatest  possible  speed,  the  useful  effect  will 
also  be  nothing,  for  then  the  animal  can  carry  no  load  ; 
and  it  becomes  a  very  useful  problem  to  determine  where 
between  these  two  limits,  the  load  and  speed  are  so  related 
that  the  useful  effect  of  the  animal  will  be  the  greatest.  By 
investigation  it  has  been  found  that  the  maximum  effect  of 
an  animal  will  be  when  it  moves  with  5  of  its  greatest 
speed,  and  carries  ^ths  of  the  greatest  load  it  can  bear. 

Thus,  if  the  greatest  speed  at  which  a  man  could  travel 
or  run,  without  a  load,  be  6  miles  per  hour ;  and  if  the 
greatest  load  which  he  can  "bear,  without  moving,  be  2£ 
cwt.,  then  this  reduced  to  Ibs.  is  280  Ibs..  hence, 

-  =  124-4  Ibs.  =  the  load,  and  — =  2  miles,  the 

y  3 

speed  with  which  a  man  will  produce  the  greatest  useful 
effect. 

Sir  John  Leslie  gives  a  formula  for  a  horse's  power,  in 
traction,  in  which  he  denotes  the  velocity  in  miles  per 
hour,  I  (12  — -  V)2  by  which  it  will  be  found  that  if  a 
horse  begins  this  pull  with  a  force  =  144  Ibs.,  he  would 
draw  100  at  the  rate  of  2  miles,  64  at  4,  and  36  at  6 ;  the 
greatest  effect  being  at  4  miles  per  hour. 

The  French  employ  a  measure  of  animal  action  which 
they  denominate  a  Dynamical  unit,  which  is  a  cubic  metre 
of  water  raised  to  the  height  of  a  metre. 

There  are  so  many  causes  operating  to  produce  variations 
in  animated  beings  even  of  the  same  kind,  that  it  is  difficult, 
if  not  impossible,  10  form  a  correct  estimate  of  the  amount 


FRICTION.  275 

of  any  one  particular  class,  or  the  comparative  strength  of 
different  classes, — hence  we  find  great  differences  in  the 
results  of  different  experimenters. 

Gregory  has  estimated  the  average  force  of  a  man  at  rest 
to  he  70  Ibs.,  and  his  utmost  walking  velocity,  when  un- 
loaded, to  be  6  feet  per  second  ;  and  that  a  man  will  pro- 
duce the  greatest  mechanical  effect  in  drawing,  when  the 
weight  was  31^ '.os.,  with  a  velocity  of  2  feet  per  second. 
But  this  is  not  the  most  advantageous  way  of  applying  the 
strength  of  men,  although  it  has  been  found  to  be  the  bes\, 
war  of  employing  the  strength  of  horses.  Robertson  Bu- 
chanan states,  that  the  mechanical  effects  of  men  in  work- 
ing a  pump,  in  turning  a  winch,  in  ringing  a  bell,  and 
rowing  a  boat,  are  as  the  numbers  100,  107,  227,  and  248. 
According  to  Hatchette,  of  a  man  working  at  the  cord  of  a 
pulley  to  raise  the  ram  of  a  pile  engine  =  50  dynamical 
units.  A  man  drawing  water  from  a  well  by  means  of  a 
cord  71  ;  a  man  working  at  a  capstan  116.  The  dynami- 
cal unit  being,  as  stated  before,  equivalent  in  English  mea- 
sure to  2208  Ibs.,  or  4  hogsheads  of  w.iter  raised  to  the 
height  of  3-281  feet  in  a  minute;  these  things  being  con- 
sidered, the  above  results  will  give  the  average  strength  of 
men  per  day. 

We  meet  with  similar  difficulties  in  estimating  the 
strength  of  horses.  According  to  Desagulicrs  and  Smea- 
ton,  1  horse  equal  to  5  men.  According  to  Bossut,  1  horse 
equal  to  7  men.  Schulze  makes  it  14  men;  and  Bossut 
states,  that  I  ass  is  equivalent  to  2  men.  It  is  also  stated 
by  Amontons,  that  2  horses  yoked  in  a  plough  exert  a 
power  of  150  Ibs. — See  the  section  on  the  Steam  Engine. 


FRICTION. 

WE  have  considered  the  effects  of  the  first  movers  of 
machinery,  and  we  must  now  direct  our  attention  to  the 
subject  of  Friction,  which,  as  we  have  frequently  noticed, 
tends  to  diminish  these  effects.  On  this  subject  it  is  not 
.OIK-  intention  to  dwell  long,  as  all  the  researches  that  have 
been  hitherto  made  in  this  branch  of  mechanical  science, 
a/e  not  of  such  a  nature  as  to  furnish  means  of  deducing 
satisfactory  laws.  The  resistance  arising  from  one  surface 


276  FRICTION. 

rubbing  against  another  is  denominated  friction  ;  and  it  is 
the  only  force  in  nature  which  is  perfectly  inert — its  ten- 
dency always  being  to  destroy  motion.  Friction  may  thus 
be  viewed  as  an  obstruction  to  the  power  of  man  in  the 
construction  of  machinery ;  but,  like  all  the  other  forces 
in  nature,  it  may,  when  properly  understood,  be  turned  to 
his  advantage, — for  friction*is  the  chief  cause  of  the  stability 
of  buildings  or  machinery,  and  without  it  animals  could  not 
exert  their  strength. 

The  friction  of  planed  woods  and  polished  metals,  with- 
out grease,  4m  one  another,  is  about  one-fourth  of  the  pres- 
sure. 

The  friction  does  not  increase  on  the  increase  of  the  rub- 
bing surfaces. 

The  friction  of  metals  is  nearly  constant. 

The  friction  of  woods  seems  to  increase  after  they  are 
some  time  in  action. 

The  friction  of  a  cylinder  rolling  down  a  plane,  is  in- 
versely as  the  diameter  of  the  cylinder. 

The  friction  of  wheels  is  as  the  diameter  of  the  axle  di- 
rectly, and  as  the  diameter  of  the  wheel  inversely.  The 
following  hints  may  be  of  use  for  the  purpose  of  diminish- 
ing friction. 

The  gudgeons  of  pivots  and  wheels  should  be  made  of 
polished  iron ;  and  the  bushes  or  collars  in  which  they 
move  should  be  made  of  polished  brass.  In  small  and 
delicate  machines,  the  pivots  or  knife  edges  should  rest  on 
garnet.  Oily  substances  diminish  friction — swine's  grease 
and  tallow  should  be  used  for  wood,  but  oil  for  metal. 
Black  lead  powder  has  been  used  with  effect  for  wooden 
gudgeons.  The  ropes  of  pulleys  should  be  rubbed  with 
tallow. 

As  to  the  friction  of  the  mechanic  powers.  The  simple 
lever  has  no  such  resistance,  unless  the  place  of  the  ful- 
crum be  moved  during  the  operation.  In  the  wheel  and 
axle  the  friction  on  the  axis  is  nearly  as  the  weight,  the 
diameter  of  the  axis,  and  the  angular  velocity — it  is,  how- 
ever, very  small.  When  the  sheaves  rub  against  the  block? 
the  friction  of  the  pulley  is  very  great.  In  most,  if  not  in 
all  screws,  the  friction  of  the  screw  is  equal  to  the  pres- 
sure— the  square  threaded  screw  is  the  best. 

In  the  inclined  plane,  the  friction  of  a  rolling  body  is 
far  less  than  that  of  a  sliding  one. 


FRICTION.  277 

To  estimate  the  amount  of  the  friction  of  a  carriage  upon 
a  railway,  we  have,  « 

P  X  T 

P =  friction, 

• 

in  which  rule  P  signifies  the  power  that  will  keep  the 
wagon  on  the  plane,  independent  of  friction,  T  the  time  of 
descent  without  friction, — both  of  which  are  to  he  deter- 
mined by  the  laws  of  the  inclined  plane  given  before  :  and 
t  is  the  time  of  actual  descent  of  the  wagon  or  carriage. 

There  is  a  loaded  carriage  on  a  railroad  120  feet  in 
length,  having  an  inclination  of  one  foot  to  the  hundred. 
The  carriage,  together  with  its  load,  weighs  4500  Ibs. 
Now,  the  height  of  the  plane  may  be  found  by  the  princi- 
ples of  geometry,  from  the  proportion  of  similar  triangles. 

IOC  :  120  :  :  1  :  i'2  ==  the  height  of  the  plane  ;  and  by 
the  laws  of  falling  bodies,  and  the  properties  of  the  inclined 
plane, 

J4?1  X  120  =  -2731  X  120  =  32-772  =  the  time 

•^(  16 

in  seconds  in  which  the  carriage  would  descend  the  plane 
without  friction — and  by  the  properties  of  the  inclined 
plane,  100  :  1  :  :  4500  :  45  =  the  force  that  sustains  the 
carriage,  without  friction,  from  rolling  down  the  plane : 
wherefore,  by  the  rule, 

45 ; 20-421    =    the   friction   in  pounds", 

60 

which  retards  the  carriage  in  rolling  down  the  railway. 


OF  MACHINES  IN  GENERAL, 

THEIR  REGULATION  AND  COMPARATIVE  EFFECTS. 


A.  MACHINE,  howsoever  complicated  it  may  be,  is  nothing 
else  than  an  organ  or  instrument  placed  between  the  work- 
men, or  source  of  force  or  power,  whatever  it  may  be,  and 
the  work  to  be  done.  Machines  are  used  chiefly  for  three 
reasons. — 1.  To  accommodate  the  direction  of  the  moving 
force  to  that  of  the  resistance  which  is  to  be  overcome.  2, 
To  render  a  power,  which  has  a  fixed  and  certain  velocity, 
effective  in  performing  work  with  a  different  velocity.  3, 
To  make  a  moving  power,  with  a  certain  intensity,  capable 
of  balancing  or  overcoming  a  resistance  of  a  greater  in- 
tensity. 

These  objects  may  be  accomplished  in  different  ways, 
either  by  using  machines  which  have  a  motion  round  some 
fixed  point,  as  the  three  first  mechanic  powers  ;  or  by  those 
which  furnish,  to  the  resistance  to  be  moved,  a  solid  path 
along  which  it  may  be  impelled,  as  is  the  case  in  the  last 
three  mechanic  powers :  hence  some  authors  have  reduced 
the  simple  machines  to  two — the  lever  and  inclined  plane. 
Simplicity  in  the  construction  of  machines  cannot  be  too 
warmly  recommended  to  the  young  engineer;  for  com- 
plexity increases  the  friction  and  expense,  and  endangers 
the  chance  of  success  from  the  derangement  of  the  parts. 
In  consequence  of  friction,  it  is  well  known,  that  no 
machine  can  overcome  a  resistance  without  an  expense  of 
the  power  of  the  first  mover,  and  as  the  more  complicated 
the  machine  is,  the  greater  will  the  friction  be  ;  so  also  will 
the  machine  be  less  powerful.  If  two  machines  be  con- 
structed, the  one  simple  and  the  other  complex,  and  be 
such,  that  the  velocity  of  the  impelled  point  is  to  the 
velocity  of  the  working  point  in  the  same  proportion  in 
both  ;  then  will  the  simple  machine  be  the  most  powerful. 

The  methods  of  communicating  motion  from  one  point 
to  another  are  infinitely  diversified  ;  and  we,  in  the  last 

278 


MACHINERY.  219 

caapter,  gave  an  account  of  the  best  of  these  which  have 
hitherto  been  invented.  We  confine  ourselves  in  the 
mean  time  to  a  few  general  remarks  on  the  construction  of 
machinery. 

When  heavy  stampers  are  to  be  raised  in  order  to  drop 
on  matter  to  be  pounded,  the  wipers  by  which  they  are 
raised  should  be  of  such  a  form,  that  the  stampers  may  be 
r.ised  by  a  uniform  pressure,  or  with  a  motion  as  nearly  as 
p.ssible  uniform.  If  this  is  not  the  case,  and  the  wiper  is 
merely  a  pin  sticking  out  of  the  axis,  the  stamper  will  be 
"•arced  into  motion  at  once,  which  will  occasion  violent  jolts 
n  the  machine,  together  with  great  strains  on  its  moving 
parts  and  points  of  support.  But  if  gradually  lifted,  no 
inequality  will  be  felt  at  the  impelled  point  of  the  machine. 
The  judicious  engineer  will  therefore  avoid,  as  much  as 
possible,  all  sudden  changes  of  motion,  especially  in  any 
ponderous  part  of  a  machine. 

When  several  stampers,  pistons,  or  other  reciprocal 
i/iovers  are  to  be  raised  and  depressed,  common  sense 
teaches  us  to  distribute  their  times  of  action  in  a  uniform 
n:anner,  so  that  the  machine  may  be  always  equally  loaded 
with  work.  When  this  is  done,  and  the  observations  in 
the  foregoing  paragraph  attended  to,  the  machine  may  be 
made  to  move  almost  as  smoothly  as  if  there  were  no  re- 
ciprocations in  it.  Nothing  shows  the  ingenuity  or  skill 
of  the  contrives  more  than  the  simple  yet  effectual  con- 
trivances for  obviating  those  difficulties  which  are  unavoid- 
able, from  the  .nature  of  the  work  to  be  done  by  the 
machine,  or  of  the  power  applied.  There  is  also  much 
ingenuity  required  in  the  management  of  the  moving 
power,  when  it  is  such  as  does  not  answer  the  kind  of  mo- 
tion required ;  for  instance,  ,n  employing  a  power  which 
necessarily  reciprocates  to  produce  a  motion  which  shall  be 
uniform,  as  in  the  employment  of  a  steam  engine  to  drive 
a  cotton  mill.  The  necessky  of  reciprocation  of  the  first 
mover  causes  a  waste  of  much  power.  The  impelling 
power  is  wasted  first  in  imparting,  and  then  in  destroying 
a  vast  quantity  of  motion  in  the  working  beam.  The  en- 
gineer will  see  the  necessity  of  erecting  the  mover  in  a 
f  separate  building  from  the  machinery  moved,  which  pre- 
vents the  great  shaking  and  speedy  destruction  of  the 
/buildings. 

The  gudgeons  of  a  water  wheel  should  ne\  er  rest  on  tho 


28C  MACHINERY 

building,  but  should  be  placed  on  a  separate  erection ;  and 
if  this  is  not  practicable,  blocks  of  oak  should  be  placed 
below  them,  which  tend  to  soften  all  tremors,  like* the 
springs  of  a  carriage. 

It  will  often  conduce  to  the  equality  of  motion  of  ma- 
chinery, to  make  the  resistance  unequal,  to  accommodate 
the  inequalities  of  the  moving  power.  There  are  some 
beautiful  specimens  of  this  kind  in  the  mechanism  of  the 
human  body. 

It  is  always  desirable,  that  the  motion  of  a  machine  should 
be  regular,  when  this  can  be  effected  ;  and  we  now  pro-* 
seed  to  state  the  various  methods  that  have  heretofore  been 
employed  for  producing  regularity  in  the  motion  of  the 
machine,  both  as  regards  the  reception  and  distribution  of 
power. 

Even  supposing  that  the  first  mover  is  perfectly  constant 
and  equable  in  its  action,  the  machine  may  not  be  regular 
in  its  movement,  from  the  irregularity  of  the  resistance  to 
be  overcome.  But  still,  if  both  the  power  and  the  resist- 
ance were  perfectly  regular,  the  machine  would  not  be 
perfectly  uniform  in  its  motion  ;  for  there  are  particular 
positions  in  which  the  moving  parts  of  a  machine  are  more 
efficacious  than  in  others,  as  in  the  crank  for  instance : 
hence  the  energy  of  the  first  mover  will  be  unequally  trans- 
mitted, and  irregularity  in  the  motion  of  the  machine  will 
consequently  follow.  The  motion  of  some  machines  bears 
a  constant  tendency  to  accelerate,  others  to  retard ;  and 
others  alternately  to  accelerate  and  retard ;  and  perhaps 
in  no  case  whatever  can  the  motion  of  a  machine  be  said  to 
be  perfectly  uniform.  But  of  this  we  will  speak  more  at 
large  when  we  come  to  treat  of  the  maximum  effect  of 
machines. 

We  intend  to  confine  our  attention  chiefly  to  the  regula- 
tors of  machinery  employed  in  the  steam  engine,  making 
occasional  remarks  on  others  as  we  go  along. 

For  the  purpose  of  regulating  the  moving  power,  the 
conical  pendulum  or  governor  is  commonly  employed. 
The  nature  of  this  beautiful  contrivance  has  been  described 
under  central  forces,  and  alluded  to  in  our  remarks  on  the 
steam  engine.  The  ring  on  the  shaft  acts  upon  a  lever  of 
the  first  kind,  whose  other  end  opens  or  shuts  a  valve,  which 
is  fixed  in  the  pipe  that  admits  the  steam  from  the  boiler  to 
the  cylinder ;  and  according  to  the  degree  of  opening  or 


MACHINERY.  281 

shutting  of  tliie  valve,  and  consequently  the  divergence  or 
convergence  of  the  halls,  or  the  velocity  of  the  shaft,  will 
be  the  quantity  of  steam  admitted  to  the  cylinder.  The 
governor  is  frequently  applied  to  the  water  wheel,  and  acts 
in  a  similar  way  by  a  board  or  valve  in  the  shuttle,  which 
delivers  the  water  to  the  wheel.  So  likewise  in  the  wind- 
mill, it  is  employed  to  furl  or  unfurl  more  or  less  sail. 

Sometimes  the  governor  is  found  inadequate  to  the  regu- 
lation of  the  machine,  and  another  contrivance  of  great 
power  and  simplicity  is  introduced.  The  machine  is  made 
to  work  a  pump,  which  tends  continually  to  fill  a  cistern 
with  water.  From  this  cistern  there  proceeds  an  eduction 
pipe,  leading  to  the  reservoir,  from  which  the  water  is 
drawn  by  the  pump.  Tliis  simple  contrivance  is  so  regu- 
lated, that  when  the  machine  goes  with  its  proper  velocity, 
the  pump  throws  just  as  much  water  into  the  cistern  as  the 
ejection  pipe  draws  from  it;  consequently,  the  water  in  the 
cistern  remains  at  the  same  level.  But  if  the  machine  goes 
too  fast,  then  the  pump  will  throw  in  more  water  than  is 
let  out  by  the  ejection  pipe,  wherefore  the  level  of  the 
water  will  rise  in  the  cistern.  If  the  machine  goes  too 
slow,  the  level  of  the  water  will  in  like  manner  fall.  Now, 
on  the  surface  of  the  water  in  the  cistern,  there  is  a  float 
which  rises  or  falls  with  the  surface  of  the  water ;  and  is 
thus  made  to  answer  the  same  purpose  as  the  ring  of  the 
governor.  It  may  be  observed,  that  the  delicacy  of  this 
kind  of  regulator  will  depend,  in  a  great  measure,  upon 
the  smallness  of  the  surface  of  the  water  which  supports  the 
float ;  for  then  a  small  difference  between  the  supply  and 
discharge,  will  cause  a  greater  difference  in  the  elevation 
or  depression  of  the  surface,  than  if  the  surface  were  large. 
To  procure  a  constant  supply  of  steam  in  the  steam  en- 
gfne,  it  is  necessary  that  the  water  in  the  boiler  be  always 
at  the  same  level.  To  effect  this  purpose,  there  is  a  lever 
fixed  on  a  support,  on  the  top  of  the  boiler,  to  one  end  of 
which  lever  there  is  attached  a  slender  rod,  which  descends 
into  the  boiler,  and  is  terminated  by  a  float,  which  rests 
on  the  surface  of  the  water  in  the  boiler.  To  the  other 
end  of  the  lever,  there  is  attached  another  rod,  to  the  end 
of  which  is  affixed  a  valve,  opening  and  shutting  the  orifice 
i  of  a  pipe  which  leads  into  the  boiler.  The  top  of  the  pipe, 
/where  the  valve  is  placed,  opens  into  a  cistern  of  water, 
is  supplied  by  a  pump  driven  by  the  engine  itself. 
24* 


282  MACIIINEUY. 

When  the  water  in  the  boiler  falls  below  its  common  level 
in  consequence  of  the  formation  of  steam,  the  float  falls 
with  it,  and  consequently  depresses  that  side  of  the  lever 
to  which  the  float  rod  is  attached  ;  the  other  arm  rises  and 
opens  the  valve  at  the  top  of  the  pipe,  which  leads  from 
the  cistern  into  the  boiler,  and  thus  admits  water  until  the 
float  rises  to  the  proper  height,  and  then  the  valve  is  closed. 
In  this  beautiful  contrivance,  the  water  is  not  supplied  to 
the  boiler  in  jolts,  but  the  float  and  valve  continuing  in  a 
state  of  constant  and  quick  vibration,  the  supply  is  rendered 
quite  constant. 

There  is  a  very  ingenious  contrivance  called  the  Tacho- 
meter, from  its  use  as  a  measure  of  small  variations  in 
velocity,  which  is  often  employed  in  the  steam  engine  and 
other  machinery.  The  simplicity  of  this  contrivance  will 
render  its  action  easily  understood.  If  a  cup  with  any 
fluid,  as.  mercury,  be  placed  on  a  spindle,  so  that  the  brim 
of  the  cup  shall  revolve  horizontally  round  its  centre,  then 
the  mercury  in  the  cup  will  assume  a  concave  form,  that  is, 
the  mercury  will  rise  on  the  sides  of  the  cup,  and  be  de- 
pressed in  the  middle ;  and  the  more  rapid  the  motion  of 
the  cup  is,  the  more  will  the  surface  of  the  mercury  differ 
from  a  plane.  Now,  if  the  mouth  of  this  cup  be  closed,  and 
a  tube  inserted  in  it,  terminated  in  the  cup  by  a  ball-shaped 
end,  and  half  filled  with  some  coloured  fluid,  as  spirits  of 
wine  and  dragon's  blood  ;  then  it  is  clear,  that  the  more 
the  surface  of  the  mercury  is  depressed,  the  more  the  fluid 
in  the  tube  will  fall,  and  vice  versa :  consequently,  the 
rapidity  or  slowness  of  the  motion  of  the  cup,  will  be  indi- 
cated by  the  height  of  the  coloured  fluid  in  the  tube  ;  and 
thus  it  becomes  a  measure  of  small  variations  in  velocity. 

In  the  steam  engine,  we  also  find  an  apparatus  for  regu- 
lating the  strength  of  the  fire  of  the  boiler,  which  apparatus 
is  called  the  self-acting  damper.  There  is  a  tube  inserted 
into  the  boiler,  reaching  nearly  to  the  bottom,  which  tube  is 
open  at  both  ends.  Now,  from  the  principles  of  Pneuma- 
tics, it  is  plain,  that  the  greater  the  pressure  of  the  steam  in 
the  boiler  is,  the  water  will  be  pressed  to  the  greater  height 
in  this  tube.  The  water  in  the  tube  supports  a  weight,  to 
which  there  is  attached  a  chain  going  over  two  wheels  ;  and 
to  the  other  end  of  the  chain  is  attached  a  plater,  which 
slides  ovei  ihe  mouth  of  the  flue  which  leads  into  the  fire. 
These  thii.gs  are  so  formed,  that  the  rising  of  the  weight 


MACHINERY.  283 

in  the  tube  will  cause  more  or  less  of  the  flue  to  he  covered 
by  the  plate  ;  and  thus  increase  or  diminish  the  quantity 
of  air  which  feeds  the  fire.  Now,  if  there  is  too  much  steam 
produced,  there  will  be  a  greater  pressure  on  the  surface 
of  the  water  in  the  boiler,  and  it  will  be  forced  up  the  tube— 
the  weight  in  the  tube  will  be  raised,  and  consequently  the 
plate  at  the  other  end  of  the  chain  will  fall,  and  cover  more 
of  the  mouth  of  the  flue,  and  thus  diminish  the  quantity 
of  air  which  feeds  the  fire ;  and  there  will  consequently  be 
generated  in  the  boiler  a  less  quantity  of  steam. 

We  come  now  to  speak  of  the  nature  and  use  of  the  fly 
wheel.  A  fly  in  mechanics  may  be  defined  to  be  a  heavy 
wheel  or  cylinder,  which  moves  rapidly  upon  its  axis,  and 
is  applied  to  a  machine  for  the  purpose  of  regulating  its 
motion. 

We  have  already  stated  that  there  are  many  circum- 
stances which  tend  to  render  the  motion  of  a  machine  ir- 
regular— variation  in  the  energy  of  the  first  mover,  whether 
it  be  water,  wind,  steam,  or  animal  strength — variation  in 
the  resistance  or  work  to  be  done — and  variations  in  the 
efficacy  of  the  machine  itself,  arising  from  the  nature  of  its 
construction,  whereby  it  is  of  necessity  more  effective  in 
one  position  than  in  another.  We  have  already  seen  how 
many  of  these  irregularities  are  compensated,  and  we  are 
now  come  to  speak  of  the  fly,  which  is  the  simplest  and 
most  effective  of  them  all.  The  principle  on  which  the  fly 
acts  is  that  of  inertia,  one  of  the  most  important  of  the  first 
principles  of  mechanical  science.  At  any  one  given  time, 
a  body  must  be  in  one  or  other  of  these  two  states — rest  or 
motion.  And  let  any  body  be  in  one  or  other  of  these  two 
state*,  it  has  no  power  within  itself  to  change  it, — if  it  be 
at  rest,  it  has  no  power  to  put  itself  in  motion — and  if  in 
motion,  it  has  no  power  in  itself  either  to  increase,  diminish, 
or  destroy  that  motion.  From  a  knowledge  of  this  fact, 
and  from  what  was  stated  before  on  the  momentum,  or 
moving  force  of  a  body,  that  it  is  the  quantity  of  matter 
multiplied  by  the  velocity  of  the  moving  body — the  nature 
of  the  operation  of  the  fly  will  be  easily  understood. 

As  the  fly  wheel,  to  do  its  office  effectually,  must  have 

a  considerable  velocity,  it  is  clear  that  its  rim,  which  has  a 

considerable  weight,  must  also  have  a  considerable  momen- 

turn,  and  consequently  a  considerable  power  to  overcome 

.  snv  tendency  either  to  increase  or  retard  its  motion. 


284  •      MACHINERY. 

To  apply  these  observations  to  actual  cases,  let  us  sup- 
pose th'at  a  single  horse  drives  a  gin.  .When  the  gin  has  been 
set  in  motion,  the  animal  cannot  exert  a  uniform  strength- 
there  will  be  occasional  increases  and  relaxations  in  the 
velocity  of  the  gin ;  but  suppose  a  fly  wheel  to  be  added, 
then,  whenever  the  animal  slackened  its  exertions,  the 
machine  would  have  a  tendency  to  move  slower,  but  the 
momentum  which  the  fly  had  acquired,  would  overcome  this 
tendency  to  retardation,  and  -continue  the  motion  of  the 
machine  at  the  same  rate  as  before,  until  the  animal  had 
recovered  iiself  so  as  to  exert  the  same  strength  as  before. 
So,  likewise,  if  the  animal  exerted  an  extraordinary  pull, 
the  inertia  of  the  wheel  would  oppose  a  resistance  which 
would  check  the  tendency  to  increase  in  the  velocity  of  the 
gin.  In  this  way  the  fly  wheel  regulates  the  motion  of  the 
gin,  whether  the  animal  takes  occasional  rests,  or  makes 
occasional  extraordinary  exertions.  It  is  evident  that  the 
fly  would  operate  in  the  same  way,  if  the  first  mover  were 
steam,  water,  or  wind,  and  that  the  other  regulators  which 
we  have  described,  are  merely  assistants  to  the  fly  wheel. 

Variations  in  the  resistance,  or  work  to  be  performed, 
are  also  compensated  by  the  fly  wheel.  For  instance,  in  a 
small  thrashing  mill  without  a  fly.  When  the  machine  is 
not  regularly  fed  with  the  corn,  there  will  be  an  occasional 
resistance,  which  will  have  a  sensible  effect  on  the  whole 
train  of  the  machinery,  even  the  water  wheel  itself;  which 
irregularity,  may,  however,  be  avoided  by  the  introduction 
of  a  fly,  as  its  inertia  will  procure  equality  of  motion :  but 
it  may  be  observed,  that  when  the  machine  is  large,  there 
will  be  less  necessity  for  a  fly,  as  the  inertia  of  the  machine 
itself  will  then  effect  the  same  purposes. 

It  was  before  stated,  that  even  supposing  the  first  mover 
and  resistance  to  be  perfectly  uniform,  the  machine  itself  is 
liable  to  variations  in  energy  at  different  positions.  It  was 
seen,  for  instance,  that  a  crank  is  more  effective  in  one 
position  than  another ;  but  the  momentum  communicated 
to  the  fly,  when  the  crank  is  in  the  most  effective  position, 
will  carry  the  crank  past  its  least  effective  position.  There 
are  many  cases,  however,  where  there  are  irregularities  of 
motion  proceeding  from  the  nature  of  the  machinery,  which 
could  be  compensated  better  than  with  a  fly.  Thus,  if  a 
bucket  is  to  be  drawn  from  the  bottom  of  a  coal  pit,  which 
is  60  fathoms  in  depth:  the  weight  of  bucket  beiug  14 


MACHINERY. 


285 


cwt.,  and  the  chain  by  which  it  is  coiled  up  round  the  cylin 
tier  weighing  8  Ibs.  to  every  fathom, — it  is  plain,  that  when 
the  bucket  is  at  the  bottom,  not  only  the  weight  of  the 
bucket,  but  also  the  weight  of  the  chain,  will  require  to  be 
overcome  in  the  raising  of  the  bucket.  Now  the  weight  of 
the  chain  is  60  +  8  =  480  Ibs.,  and  the  amount  of  the 
weight  of  the  bucket  is  14  cwt.  or  1568  Ibs. ;  hence  1  568  + 
480  =  2048  Ibs. ;  but  the  weight  of  the  chain  will  always 
be  getting  less  as  it  is  coiled  round  the  cylinder,  until  the 
bucket  comes  to  the  cylinder,  when  the  chain  will  be  all 
coiled,  and  there  will  remain  only  the  weight  of  the  bucket. 
Now,  the  use  of  a  fly  may  be  advantageously  dispensed 
with,  if  the  barrel  on  which  the  chain  is  coiled  is  formed 
like  a  cone ;  the  diameter  of  the  barrel  thus  increasing  with 
the  uniform  diminution  of  the  weight. 

The  effect  of  the  fly  wheel  in  accumulating  force,  has  led 
some  to  suppose  that  there  is,  positively,  a  creation  of  force 
in  the  fly ;  but  this  is  a  mistake,  for  it  is  only,  as  it  were,  a 
magazine  of  power,  where  there  is  no  force  but  what  has 
been  delivered  to  it.  The  great  use  of  the  fly  wheel  is  thus 
to  deliver  out  at  proper  intervals,  that  force  which  has  been 
previously  communicated  to  it ;  and  although  there  is  ab- 
solutely a  small  loss  of  power  by  the  use  of  the  fly,  yet  this 
is  more  than  compensated  by  its  utility  as  a  regulator. 

The  motion  of  machines  may,  as  stated  before,  be  re- 
duced to  three  kinds.  That  which  is  gradually  accelerated, 
which  generally  takes  place  at  the  commencement  of  a 
machine's  action:  that  which  is  entirely  uniform:  that 
which  is  alternately  accelerated  and  retarded.  The  nearer 
that  the  motion  of  a  machine  approaches  to  uniformity,  the 
greater  will  he  the  quantity  of  work  done. 

In  order  that  the  few  remarks,  which  we  intend  to  make 
on  the  effect  of  machines,  may  be  clearly  understood,  we 
desire  the  reader  to  attend  to  the  following  definitions. 

The  impelled  point  of  any  machine,  is  that  point  at  which 
the  force  which  moves  the  machine,  may  be  considered  as 
applied — as  the  piston  of  a  steam  engine,  or  the  float  board 
of  a  water  wheel. 

The  working  point,  on  the  contrary,  is  that  point  where 
the  resistance  may  be  supposed  to  act. 

The  velocity  of  the  moving  power  is  the  same  as  the  ve- 
locity of  the  impelled  point, — the  velocity  of  the  resistance 
is  the  same  as  the  velocity  of  the  working  point. 


886  MACHINERY. 

The  performance  or  effect  of  a  machine  is  measured  by 
the  resistance  or  work  performed,  (calculated  by  weight,) 
multiplied  by  its  velocity,  which  is,  in  other  words,  the  mo- 
mentum of  the  working  point.  The  momentum  of  impulse, 
on  the  other  hand,  is  measured  by  the  energy  of  the  first 
mover,  (also  estimated  by  weight,)  multiplied  by  the  ve- 
locity of  the  impelled  point. 

These  definitions  being  understood,  we  proceed  to  a 
simple  statement  of  principles. 

When  any  power  is  made  to  act  in  opposition  to  a  resist- 
ance, by  means  either  of  a  simple  or  compound  machine  ; 
which  machine  will  be  in  a  state  of  rest,  when  the  velocity 
of  the  power  is  to  that  of  the  resistance  as  the  weight  of 
the  resistance  is  to  that  of  the  power.  In  this  state  of  things 
the  machine  can  do  no  work,  because  it  has  no  motion ;  but 
if  the  power  is  increased,  so  as  to  overcome  the  resistance, 
the  machine  will  have  an  accelerated  motion  so  long  as 
the  power  exceeds  the  resistance.  If  the  power  should 
diminish,  the  machine  would  accelerate  less  and  less,  until 
its  motion  became  uniform.  The  same  effect  would  ne- 
cessarily follow,  if  the  resistance  increased,  a  circumstance 
which  may  arise  from  various  causes.  From  the  resistance 
of  the  air,  which  increases  with  an  increase  of  velocity ; 
and  also  from  friction,  which  often  increases  with  the  in- 
crease of  velocity.  Hence  we  find,  that  machines  have 
commonly  a  tendency  to  become  uniform  in  their  motion. 

We  have  seen  before,  while  treating  of  the  water  wheel, 
that  the  velocity  of  the  floats  of  the  undershot  wheel,  must 
be  less  than  the  velocity  of  the  stream.  For,  when  the 
float  board  is  at  rest,  the  water  will  impinge  on  it  with  the 
greatest  possible  effect ;  but  so  soon  as  the  float  begins  to 
move,  then  it  leaves  the  water,  as  it  were,  and  does  not  re- 
ceive the  whole  impetus  of  the  stream  ;  and  if  the  velocity 
of  the  float  were  equal  to  that  of  the  stream,  it  is  clear  that 
the  water  would  have  no  effect  upon  it  at  all ;  and,  as  was 
stated  before,  there  is  a  certain  relation  between  the  velocity 
of  the  wheel  and  that  of  the  stream,  at  which  the  effect  will 
be  a  maximum.  This  is  not  confined  to  the  water  wheel, 
but  is  common  to  all  machines,  as  we  have  seen  illustrated 
in  the  steam  engine. 

We  have  seen  before,  that  the  maximum  effect  of  an 
animal  was,  when  its  velocity  was  one-third  of  its  creates 
possible  speed,  and  the  load  which  it  bore,  or  the  resistance 


MACHINERY.  '287 

which  it  overcame,  was  equal  to  four-ninths  of  its  greatest 
possible  load. 

The  following  tables  (A  and  B)  constructed  from  the  re- 
sults of  Dr.  Kobison,  will  be  useful  to  the  mechanic. 

Table  A  contains  the  least  proportions  between  the  velo- 
cities of  the  impelled  and  working  points  of  a  machine ;  or 
between  the  levers  by  which  the  power  and  resistance  act. 

The  use  of  this  table  is  very  simple,  for  suppose  we 
wished  to  raise  3  cubic  feet  of  water  per  second,  by  means 
of  a  water  wheel,  whose  radius  was  8  feet,  (=  the  length 
of  the  lever  by  which  the  power  acts,)  and  the  power  which 
moves  the  wheel  being  6  cubic  feet  of  water  per  second. 

Employ  this  rule  : 

Power, 

— — : x  10  =  a  number, 

Resistance, 

which  look  for  in  column  M,  and  against  it  in  column  N, 
will  be  found  a  number  which,  when  multiplied  by  the 
length  of  lever  at  which  the  power  acts,  will  give  the  lengtK 
of  lever  at  which  the  resistance  should  act. 

Wherefore,  in  the  above  case, 
/» 

—  X  10  =  20,  the  number  corresponding  to  which  is 

0-732051,  hence  0-732051  X  8  =  5-856408  =  the  radius 
of  the  axle  at  which  the  resistance  or  work  to  be  done  acts. 
This  table  will  be  found  very  useful  in  the  construction 
of  machines  ;  but  they  are  frequently  already  constructed, 
and  it  becomes  then  necessary  for  us  to  regulate  the  power 
and  resistance  in  order  to  produce  a  maximum  effect,  with- 
out making  -\ny  alteration  in  the  machine.  For  this  pur- 
pose we  employ  table  B,  in  order  to  show  the  use  of  which 
*re  give  the  following  rule  and  example  : 

Length  of  lever  of  resistance, 

— T-J —  =  a  number,  which,  when 

Length  of  lever  of  power, 

found  in  column  O,  will  stand  against  a  number  in  column 
P:  such,  when  multiplied  by  the  energy  of  power,  will  give 
the  proper  energy  of  resistance.  Thus,  if  a  man  exerts  a 
constant  force  of  56  Ibs.  on  the  handle  of  a  capstan,  whose 
leverage  is  4  feet,  and  the  barrel  is  one  foot  in  radius,  then 
we  have. 

=•-  — a  number,  which  will  be  found  in  column  O,  cor 
4        4 


288 


MACHINERY. 


Desponding  to  which  will  be  found,  in  column  P,  the  num- 
ber 1*8885  ;  wherefore,  by  the  rule, 

1-8885  x  56  =  105-756  =  the  resistance  which  the  man 
in  these  circumstances,  can  overcome  with  the  greatest  ad- 
vantage, or  with  the  maximum  mechanical  effect. 

TABLE  A. 


TABLE  B. 


M 

N 

M 

N 

1 

0-048809 

20 

0-732051 

2 

0-095445 

21 

0-760682 

3 

0-140175  - 

22 

0-788854 

4 

0-183216 

23 

0-816590 

5 

0-224745 

24 

0-843900 

6 

0-264911 

25 

0-870800 

7 

0-303841 

26 

0-897300 

8 

0-341641 

27 

0-923500 

9 

0-378405 

28 

0-949400 

10 

0-414211 

29 

.  0-974800 

11 

0-449138 

30 

1-000000 

12 

0-483240 

40 

1-236200 

13 

0-516575 

50 

1-449500 

•14 

0-549193 

60 

1-645600 

15 

0-581139 

70 

1-828400 

16 

0-612451 

80 

2-200000 

17 

0-643168 

90 

2-162300 

18 

0-673320 

100 

2-.3  16600 

19 

0-702938 

o 

P 

0 

P 

* 

1-8885 

7 

0-03731 

1 

3 

1-3928 

8 

0-03125 

3 

0-8986 

9 

0-02669 

1 

0-4142 

10 

0-02317 

2 

0-1830 

11 

0-02037 

3 

0-1111 

12 

0-01809 

4 

0-0772 

13 

0-01622 

5 

0-0587 

14 

0-01466 

6 

0-0457 

15 

0-01333 

MACHINERY. 


28S 


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290  COTTON    SPINNING. 

It  is  not  by  any  means  an  easy  matter  to  estimate  the 
relative  quantities  of  work  done  by  different  machines. 
Their  effects  are  generally  stated  as  equivalent  to  so  many 
horses'  power,  and  the  following  data  are  commonly  given: 
One  horse's  power,  at  a  maximum,  is  equivalent  to  the 
raising  of  1000  Ibs.  13  feet  high  in  one  minute.  In  cotton 
factories,  100  spindles,  with  preparation,  are  allowed  to  each 
horse  power  for  spinning  cotton  yarn  twist,  or  five  time? 
that  number  of  spindles,  with  preparation,  for  mule  yarn, 
No.  48  ;  and  if  it  be  No.  110,  ten  times  that  number  of 
spindles,  with  preparation — and  the  power-loom  factories 
12  beams  with  subservient  machinery. 

Thus  a  steam  engine  on  Watt's  principle,  having  a  cylin- 
der of  30  inches  diameter,  and  a  stroke  of  6  feet,  making 
21  double  strokes  per  minute,  will  give,  by  the  usual  cal- 
culation, 

•7854  X  30g  X  10  X  6  X  21  X  2  _ 

44000 

40  horses'  power.  Hence  such  an  engine  will  drive  4000 
spindles  cotton  yarn  twist,  or  20,000  spindles  mule  twist, 
No.  48,  or  40,000  mule  twist  spindles,  No.  110,  or  480 
power-looms — in  each  of  which  cases  subservient  or  pre 
paratory  machinery  is  included. 


RULES   FOR  COTTON  SPINNERS. 

IN  the  following  calculations  the  reader  is  supposed  to  be 
acquainted  with  the  construction  of  the  various  machines 
employed  in  the  cotton  manufacture,  so  that  the  rules  are 
only  intended  to  assist  the  memory  of  the  practical  man  in 
cases  of  particular  difficulty.  The  effects  of  shafts,  belts, 
drums,  pulleys,  pinions,  and  wheels,  in  varying  velocity, 
depend  upon  the  principles  established  when  treating  of  the 
mechanical  powers,  and  the  calculations  connected  with 
them  may  be  easily  performed  by  the  rules  given  in  that 
section. 

To  find  the  draught  on  the  spreading  machine,  count  the 
number  of  teeth  of  the  wheel  on  the  end  of  the  feeding  roller 
shaft,  calling  it  the  first  leader,  and  also  the  number  of  teeth 
on  the  pinion  which  it  drives,  calling  it  the  first  follower, 
and  in  like  manner  reckon  all  the  leaders  and  followers  on 


COTTON    SPINNING.  291 

to  the  last  follower  i.  e.  the  wheel  on  the  calender  roller 
shaft,  omitting  all  intermediate  wheels,  then, 

product  of  leaders  x  diam.  calender  roller 
product  of  followers  x  feeding  roller 

If  the  teeth  of  the  leaders  be  160,  22,  and  20,  and  those 
of  the  followers  90,  22  and  40 ;  the  diameter  of  calender 
roller  5,  and  feeding  roller  2  inches  ;  then, 

160  X22  X20X5 

-  =  2-26  =  the  draught. 
90  x  22  x  40  x  2 

The  reader  will  have  no  difficulty  in  applying  the  prin- 
ciple of  this  rule  to  the  calculation  of  the  draught  of  other 
machines  in  cotton  manufacture. 

To  find  the  number  of  twists  per  inch  given  to  the  rove 
by  the  fly  frame  : — 

Turns  of  front  roller  per  minute  x  its  circumference  = 
length  of  rove  produced  in  one  minute,  dividing  the  turns 
of  the  spindle  per  minute  by  that  product,  gives  the  number 
of  twists  on  the  rove  per  inch. 

Let  the  revolutions  of  the  front  roller  per  minute  be  100, 
and  the  circumference  4  inches,  then  100  x  4  =  400  inches 
=  33  feet  4  inches  of  rove  produced  in  a  minute,  where- 
fore, if  the  spindle  revolve  600  times  in  a  minute,  then, 

600 

=  1'5  twists  per  inch. 

The  proper  diameter  of  the  taking-out  pulley,  or  men- 
doza  pulley  of  the  stretching  frame  that  shall  regulate  the 
motion  of  the  carriage  to  the  delivery  of  the  rove,  may  be 
found  by  taking  the  product  of  the  diameter  of  the  front 
roller  X  the  number  of  teeth  in  the  mendoza  wheel,  and 
dividing  by  the  number  of  teeth  in  the  front  roller  pinion, 
and  subtracting  from  the  quotient  the  diameter  of  the  men- 
doza bond.  Thus  if  the  diameter  of  the  front  roller  be  1| 
inches,  the  diameter  of  the  mendoza  bond  5  inch,  the  teeth 
in  front  roller  pinion  20,  and  in  mendoza  wheel  110,  then, 

110  x  U  137-5 

i  —  i  =  _ I  =  6-8  —  -5  =  6-3  inches 

<£0  *U 

=  the  diameter  of  mendoza  pulley. 

/The  revolutions  of  the  spindle  of  the  throstle  may  be 
found  thus  : — 


292  COTTOTS    SPINNING. 

turns  of  cylinder  per  minute  x  its  diameter 
diameter  of  wharve 

A  cylinder  of  7*5  inches  diameter  makes  450  revolutiona 
per  minute,  and  the  diameter  of  the  wharve  is  1  inch, 
nence, 

450  X  7'5 

-  -  -  =  3375  =  turns  of  the  spindle  per  minute. 

To  find  the  draught  of  the  roller  of  the  jenny,  take  the 
product  of  the  teeth  of  the  front  roller  pinion  X  the  grist 
pinion  X  diameter  of  back  roller  for  a  divisor,  and  take 
the  product  of  the  diameter  of  front  roller  X  the  number  of 
teeth  of  the  crown  wheel  x  those  of  the  back  roller  wheel 
for  a  dividend,  then  the  dividend  divided  by  the  divisor  will 
give  the  draught.  Thus  if  the  teeth  of  the  crown  wheel  be 
72,  back  roller  wheel  56,  front  roller  pinion  18,  and  grist 
pinion  24,  the  diameter  of  front  roller  1  inch,  and  of  back 
roller  |,  then, 

72  x  56  x  1 


18X24X| 

In  order  to  determine  the  size  of  yarn  from  hank  rove, 
we  must  first  find  the  quantity  of  rove  given  out  by  the 
roller  during  one  stretch,  which  is  =  the  whole  length  of 
stretch  —  the  inches  gained,  and  calling  this  the  divisor, 
the  dividend  will  be  found  by  taking  the  product  of  the 
number  of  hank  rove  X  the  length  of  the  stretch  X  the 
draught,  the  quotient  will  be  the  size  of  yarn  produced. 
Thus,  if  the  draught  be  as  found  above  10*666,  the  stretch 
56,  the  gaining  of  carriage  5  inches,  and  the  rove  5  hank, 
then, 

10-66  X  5  x  56  r 

-  —  -  -  -  -  =  58-52  =  size  of  yarn. 
56  —  5 

To  find  the  effect  of  a  change  of  the  grist  pinion  on  the 
jenny. 

Take  the  product  of  the  pinion  producing  a  known  size 
of  yarn,  and  call  it  the  dividend,  if  this  be  divided  by  any 
other  number  of  yarn,  the  quotient  will  be  the  correspond- 
ing grist  pinion  ;  or  if  another  grist  pinion  be  used  as  a 
divisor,  the  quotient  will  be  the  corresponding  size  of  yarn 
produced.  Thus  if  No.  70  yarn  be  produced  by  a  pinion 
of  24  teeth,  then, 


COTTOP;    SPINNING.  293 

24  x  7- 

-  =  28  =  the  number  of  teeth  in  a  grist  pinion 

that  shall  produce  yarn  No.  60 ;  and  also 

24  x  70 

— — —  =  42  =  the  number  of  yarn  that  shall  be  pro- 
duced by  a  grist  pinion  of  42  teeth. 

Take  the  product  of  the  diameter  of  the  front  roller  X 
the  teeth  of  the  mendoza  wheel,  and  divide  by  the  teeth  of 
the  pinion  on  the  front  roller  that  drives  the  mendoza 
wheel.  From  the  quotient  thus  found,  subtract  the  diame- 
ter of  the  mendoza  band,  and  the  remainder  is  the  diameter 
of  a  pulley  that  will  move  the  carriage  out  with  the  same 
speed  as  the  yarn  passes  through  the  front  rollers.  When 
this  is  found,  the  diameter  of  such  a  pinion  as  will  give  a 
certain  gain  on  the  stretch  may  be  found  by  multiplying  " 
the  last  result  by  the  full  length  of  the  stretch,  and  divide  : 
the  product  bv  the  difference  of  the  length  of  the  stretch 
and  the  gaining  required.*  Thus,  if  the  length  of  stretch 
be  56  inches,  the  gain  upon  stretch  5  inches,  the  diameter 
of  the  front  roller  1  inch,  and  of  the  mendoza  band  f-  of  an 
inch,  the  number  of  teeth  on  the  mendoza  wheel  112,  and 
on  the  front  roller  pinion  18,  then, 

^-  =  6-22  —  -625  =  5-595  =  the 

diameter  of  mendoza  pulley,  to  move  the  carriage  uniformly 
with  the  delivery  of  the  front  roller,  and 

56X5-595       313-32 

— — -—  =  — — —  =  6-14  =  the  diameter  of  men 

56  —  5  51 

doza  pulley  to  move  the  carriage  with  a  gain  of  five  inche* 
on  the  stretch. 

The  number  of  twists  given  to  cotton  yarn  varies  with 
the  quality  of  the  fibre  of  the  wool,  the  fineness  of  the  yarn, 
and  whether  it  be  intended  for  warp  or  weft.  But  omitting 
the  variation  necessary  for  difference  in  the  length  of  fibre, 
which  is  comparatively  trifling,  the  number  of  twists  in  the 
inch  will  vary  with  the  square  root  of  the  No.  of  the  yarn, 
or  a  go<^l  practical  rule  is  this, 

x/No.  of  yarn  x  3-75  for  the  twists  per  inch  cf  warp 
yarn,  and 

v/No.  of  yarn  X  3-25  for  wefts. 
25* 


294  COTTON    SPINNING 

Th^s  for  No.  36  warps,  we  have, 
v/36  X  3-75  =  6  x  3-75  =  22'5  twists  per  inch 
And  for  No.  64  wefts, 

v/64  X  3-25  =  8  X  3'25  =  26  twists  per  inch. 
When  cotton  yarn  is  put  up  in  hanks  or  spindles,  it  is 
coiled  upon'  a  reel,  one  revolution  of  which  takes  up  54 
inches  of  thread,  and  this  length  of  yarn  is  denominated  a 
thread. 

54  in.  =   lj  yards  =  1  thread  or  round  of  the  reel. 

120  =         80  =  1  skein  or  ley. 

840  =       560  =  7  ==  1  hank  or  No. 

15120  =  10080  =  126  =  18  =  1  spindle. 

Cotton  yarn  is  sold  by  weight,  and  its  fineness  is  esti- 
mated by  the  No.  of  hanks  in  a  pound.  Thus,  No.  20 
yarn  contains  20  hanks,  or  20  X  840  yards  =  16800  yards 
in  one  pound ;  No.  64  contains  64  hanks  or  64  x  840  = 
53760  yards  of  thread  in  a  pound ;  consequently  the 
diameter  of  the  thread  of  No.  64  must  be  much  less  than 
the  diameter  of  the  thread  of  No.^0. 

When  the  yarn  is  in  cops  the  fineness  is  determined  by 
reeling  a  few  hanks,  and  by  finding  their  weight,  the  No. 
of  the  yarn  may  be  found  by  proportion ;  thus  if  a  spindle 
be  reeled,  and  its  weight  found  to  be  4  ounces  8  drachms, 
then  by  proportion,  since  there  are  18  hanks  in  a  spindle, 
and  16  ounces  in  a  pound,  and  16  drachms  in  an  ounce, 
we  have, 

4£  :  16  ::  18  :  64  =  the  number  of  the  yarn;  or, 

288 

— ^-r — ? ..    . =  No.  of  yarn ; 

weight  of  a  spindle  in  oz. 

and, 

288 
-=^ j SB  weight  of  a  spindle  in  ounce*. 


SQUARE    AND    CUBE    ROOTS. 


295 


'  Mix 

Square  root 

Cube  net 

V 

Square  root. 

Cube  root. 

No. 

Squire  root. 

Cot*  root. 

1 

1- 

1- 

49 

7- 

3-659 

97 

9-8488 

4.594 

2 

1-4142 

.1-259 

50 

7-0710 

3-684 

9S 

9-8994 

4-610 

3 

1-7320 

1-442 

7-1414 

3-708 

99 

9-9498 

4-626 

4 

2- 

1-587 

52 

7-2111 

3-732 

100 

10- 

4-641 

5 

2-2360 

1-709  ||  53 

7-2801 

3-756 

101 

10-0498 

4-657 

6 

2-4494 

1-817 

54 

7-3484 

3-779 

102 

10-0995 

4-672 

7 

2-6457 

1-912 

55 

7-4161 

3-802 

103 

10-1488 

4-687 

8 

2-8284 

2- 

56 

7-4833 

3-825 

104 

10-1980 

4-702 

9 

3- 

2-080 

67 

7-5498 

3-848 

105 

10-2469 

4-717 

10 

3-1023 

2-154 

58 

7-6157 

3-870 

106 

10-2956 

4-732 

11 

3-3  HiG 

2-223 

59 

7-0811 

3-892 

107 

10-3440 

4-747 

12 

3-4  (141 

2-289 

60 

7-7459 

8-914 

108 

10-3923 

4-762 

13 

3-C055 

2-351 

61 

7-8102 

3-936 

109 

10-4403 

4-776 

14 

3-7416 

2-410 

62 

7-8740 

3-957 

110 

10-4880 

4-791 

15 

3-8729 

2-466 

63 

7-9372  3-979 

111 

10-5356 

4-805 

16 

4- 

2-519 

64 

8- 

4- 

112 

10-5830 

4-820 

17 

4-1231 

2-571 

65 

8-0622 

4-020 

113 

10-6301 

4-834 

18 

4-2426 

2-620 

66 

8-1240 

4-041 

114 

10-6770 

4-848 

19 

4-3588 

2-668 

67 

8-1853 

4-061 

115 

10-7238 

4-862 

20 

4-4721 

2-714 

68 

8-2462 

4-081 

116 

10-7703 

4-876 

21 

4-5825 

2-758 

69 

8-3066 

4-101 

117 

10-8166 

4-890 

22 

4-6904 

2-802 

70 

8-3666 

4-121 

118 

10-8627 

4-904 

23 

4-7958 

2-843 

71 

8-4261 

4-140 

119 

10-9087 

4-918 

24 

4-8989 

2-884 

72 

8-4852 

4-160 

120 

10-9544 

4-932 

M 

5- 

2-924 

73 

8-5440 

4-179 

121 

11- 

4-946 

26 

5-0990 

2-962 

74 

8-6023 

4-198 

122 

11-0453 

4-959 

27 

5-1961 

3- 

75 

8-6602 

4-217 

123 

11-0905 

4-973 

28 

5-2915 

3-036 

76 

8-7177 

4-235 

124 

11-1355 

4-986 

29 

5-3851 

3-072 

77 

8-7749 

4-254 

125 

11-1803 

5- 

30 

5-4772 

3-107 

78 

8-8317 

4-272 

126 

11-2249 

5-013 

31 

5-5677 

3-141 

79 

8-8881 

4-290 

127 

11-2694 

5-026 

32 

5-6568 

3-174 

80 

8-9442 

4-308 

128 

11-3137 

5-029 

33 

0-7445 

3-207 

81 

9- 

4-326 

129 

11-3578 

5-052 

S4 

5-8309 

3-239 

82 

9-0553 

4-344 

130 

11-4017 

5-065 

3.-> 

5-9160 

3-271 

83 

9-1104 

4-362 

131 

11-4455 

5-078 

96 

6- 

3-301 

84 

9-1651 

4-379 

132 

11-4891 

5-091 

37 

6-0827 

3-332 

85 

9-2195 

4-396 

133 

11-5325 

5-104 

38 

6-1644 

3-361 

86 

9-2736 

4-414 

134 

11-5758 

5-117 

39  G-2449 

3-391 

9-3-273 

4-431 

135 

11-6189 

5-129 

40  6-3245 

8419 

9-3808  4-447 

136 

11-6019 

5-142 

41  6-4031 

3-448 

89  9-4339  4-464 

137 

11-7046 

5-155 

42  6-4807 

3-476  |  90  9-4868  4-481 

138i  11-7473:5-167 

43  6-5574 

44  (;•<;:<:••.: 

3-503  :  91  1  9-5393  ;  4-497 
3530   92  9-5916,4-514 

139  11--, 
140  ll-83vil 

5-180 
5-192 

45 

6-7082 

3-556 

93 

9-043I)  4-530 

141  11-8743 

5-204 

46 

6-7823 

3583 

94  9-U953  4-546 

142 

119163 

5-217 

47  6-8556 

3608 

95 

9-7467  4-562 

143 

11-9582 

5-229 

48  6-9282 

3-634   96 

9-7979 

4-578 

144 

12- 

5-241 

25* 


296 


SQUARE    AJCD    CUBE    ROOTS. 


No. 

Square  root. 

Cube  root. 

HO. 

Square  root. 

Cube  root. 

No. 

Square  root. 

Cab*  root 

145 

12-0415 

5-253 

193 

13-8924 

5-778 

241 

15-5241 

6-223 

146 

12-0830 

5-265 

194 

13-9283 

5-788 

242 

15-5563 

6-231 

147 

12-1243 

5-277 

195 

13-9642 

5-798 

243 

15-5884 

6-240 

148 

12-1655 

5-289 

196 

14- 

5-808, 

244 

15-6204 

6-248 

149 

12-2065 

5-301 

197 

14-0356 

5-818 

245 

15-6524 

6-257 

150 

12-2474 

5-313 

198 

14-0712 

5-828 

246 

15-6843 

6-265 

151 

12-2882 

5-325 

199 

14-1067 

5-838 

247 

15-7162 

6-274 

152 

12-3288 

5-336 

200 

14-1421 

5-848 

248 

15-7480 

6-282 

153 

12-3693 

5-348 

201 

14-1774 

5-857 

249 

15-7797 

6-291 

154 

12-4096 

5-360 

202 

14-2126 

5-867 

250 

15-8113 

6-299 

155 

12-4498 

5-371 

203 

14-2478 

5-877 

251 

15-8429 

6-307 

156 

12-4899 

5-383 

204 

14-2828 

5-886 

252 

15-8745 

6-316 

157 

12-5299 

5-394 

205 

14-3178 

5-896 

253 

15-9059 

6-324 

158 

12-5698 

5-406 

206 

14-3527 

5-905 

254 

15-9373 

f  333 

159 

12-6095 

5-417 

207 

14-3874 

5-915 

255 

15-9687 

&-341 

160 

12-6491 

5-428 

208 

14-4222 

5-924 

256 

16- 

6-349 

161 

12-6885 

5-440 

209 

14-4568 

5-934 

257 

16-0312 

6-357 

162 

12-7279 

5-451 

210 

14-4913 

5-943 

258 

16-0623 

6-366 

163 

12-7671 

5-462 

211 

14-5258 

5-953 

259 

16-0934 

6-374 

164 

12-8062 

5-473 

212 

14-5602 

5-962 

260 

16-1245 

6«382 

165 

12-8452 

5-484 

213 

14-5945 

5-972 

261 

16-1554 

6-390 

166 

12-8840 

5-495 

214 

14-6287 

5-981 

262 

16-1864 

6-398 

167 

12-9228 

5-506 

215 

14-6628 

5-990 

263 

16-2172 

6-406 

168 

12-9614 

5-517 

216 

14-6969 

6- 

264 

16-2480 

6-415 

169 

13- 

5-528 

217 

14-7309 

6-009 

265 

16-2788 

6-423 

170 

13-0384 

5-539 

218 

14-7648 

6-018 

266 

16-3095 

6-431 

171 

13-0766 

5-550 

219 

14-7986 

6-027 

267 

16-3401 

6-439 

172 

13-1148 

5-561 

220 

14-8323 

6-036 

268 

16-3707 

6-447 

173 

13-1529 

5-572 

221 

14-8660 

6-045 

269 

16-4012 

6-455 

174 

13-1909 

5-582 

222 

14-8996 

6-055 

270 

16-4316 

6-463 

175 

13-2287 

5-593 

223 

14-9331 

6-064 

271 

16-4620 

6-471 

176 

13-2664 

5-604 

224 

14-9666 

6-073 

272 

16-4924 

6-479 

177 

13-3041 

5-614 

225 

15- 

6-082 

273 

16-5227 

6-487 

178 

13-3416 

5-625 

226 

15-0332 

6-091 

274 

16-5529 

6-495 

179 

13-3790 

5-635 

227 

15-0665 

6-100 

275 

16-5831 

6-502 

180 

13-4164 

5-646 

228 

15-0996 

6-109 

276 

16-6132 

6-510 

181 

13-4536 

5-656 

229 

15-1327 

6-118 

277 

16-6433 

6-518 

182 

13-4907 

5-667 

230 

15-1657 

6-126 

278 

16-6733 

6-526 

183 

13-5277 

5-677 

231 

15-1986 

6-135 

279 

16-7032 

6-534 

184 

13-5646 

5-687 

232 

15-2315 

6-144 

280 

16-7332 

6-542 

185 

13-6014 

5-698 

233 

15-2643 

6-153 

281 

16-7630 

6-549 

186 

13-6381 

5-708 

234 

15-2970 

6-162 

282 

16-7928 

6-557 

187 

13-6747 

5-718 

235 

15-3297 

6-171 

283 

16-8226 

6-565 

188 

13-7113 

5-728 

236 

15-3622 

6-179 

284 

16-8522 

6-573 

189 

13-7477 

5-738 

237 

15-3948 

6-188 

285 

16-8819 

6-580 

190 

13-7840 

5-748 

238 

15-4272 

6-197 

286 

16-9115 

6-588 

191 

13-8202 

5-758 

239 

15-4596 

6-205 

287 

16-9410 

6-596 

192 

13-8564 

5-768 

240 

15-4919 

6-214 

288 

16-9705 

6-603 

SQUARE    AND    CUBE    ROOTS. 


297 


;: 

8quu«rool. 

'ubr  root. 

No. 

Square  root. 

?ute  root. 

No. 

Square  root. 

Cub*  roc  1 

289 

17- 

6-611 

337 

18-3575 

6-958 

385 

19-6214 

7-274 

290 

17-0293 

6-<>19 

338 

18-3847 

6-965 

386 

19-6468 

7-281 

291 

17-0587 

6-626 

339 

18-4119 

6-972 

387 

19-6723 

7-287 

292 

17-0880 

6-634 

340 

18-4390 

6-979 

388 

19-6977 

7-293 

293 

17-1172 

6-641 

341 

18-4661 

6-986 

389 

19-7230 

7-299 

294 

17-1464 

6-649 

342 

18-4932 

6-993 

390 

19-7484 

7-306 

295 

17-1755 

6-656 

343 

18-5202 

7- 

391 

19-7737 

7-312 

296 

17-2046 

6-664 

344 

18-5472 

7-006 

392 

19-7989 

7318 

297 

17-2336 

6-671 

345 

18-5741 

7-013 

393 

19-8242 

7-324 

298 

17-2626 

6-679 

346 

18-6010 

7-020 

394 

19-8494 

7-331 

299 

17-2916 

6-686 

347 

18-6279 

7-027 

395 

19-8746 

7-337 

300 

17-3205 

6-694 

348 

18-6547 

7-033 

396 

19-8997 

7-343 

301 

17-3493 

6-701 

349 

18-6815 

7-040  : 

397 

19-9248 

7-349 

302 

17-3781 

6-709 

350 

18-7082 

7-047 

398 

19-9499 

7-355 

303 

17-4068 

6-716 

351 

18-7349 

7-054 

399 

19-9749 

7-361 

304 

17-4355 

6-723 

352 

18-7616 

7-060 

400 

20- 

7-368 

305 

17-4642 

6-731 

353 

18-7882 

7-067 

401 

20-0249 

7-374 

306 

17-4928 

6-738 

354 

18-8148 

7-074 

402 

20-0499 

7-380 

307 

17-5214 

6-745 

355 

18-8414 

7-080 

403 

20-0748 

7-386 

308 

17-5499 

6-753 

356 

18-8679 

7-087 

404 

20-0997 

7-39S 

309 

17-5783 

6-760 

357 

18-8944 

7-093 

405 

20-1246 

7-398 

310 

17-6068 

6-767 

358 

18-9208 

7-100 

406 

20-1494 

7-404 

311 

17-6351 

6-775 

359 

18-9472 

7-107 

407 

20-1742 

7-410 

312 

17-6635 

6-782 

360 

18-9736 

7-113 

408 

20-1990 

7-416 

313 

17-6918 

6-789 

361 

19- 

7-120 

409 

20-2237 

7-422 

314 

17-7200 

6-796 

362 

19-0262 

7-126 

410 

20-2484 

7-428 

315 

17-7482 

6-804 

363 

19-0525 

7-133 

411 

20-2731 

7-434 

316 

17-7763 

6-811 

364 

19-0787 

7-140 

412 

20-2977 

7-441 

317 

17-8044 

6-818 

365 

19-1049 

7-146 

413 

20-3224 

7-447 

318 

17-8325 

6-825 

366 

19-1311 

7-153 

414 

20-3469 

7-453 

319 

17-8605 

6-832 

367 

19-1572 

7-159 

415 

20-3715 

7-459 

320 

17-8885 

6-839 

368 

19-1833 

7-166 

416 

20-3960 

7-465 

321 

17-9164 

6-847 

369 

19-2093 

7-172 

417 

20-4205 

7-470 

322 

17-9443 

6-854 

370 

19-2353 

7-179 

418 

20-4450 

7-476 

323 

17-9722 

6-861' 

371 

19-2613 

7-185 

419 

20-4694 

7-482 

324 

18- 

6-868 

372 

19-2873 

7-191 

420 

20-4939 

7-488 

325 

18-0277 

6-875 

373 

19-3132 

7-198 

421 

20-5182 

7-494 

326 

18-0554 

6-882 

374 

19-3390 

7-204 

422 

20-5426 

7-5dO 

327 

18-0831 

6-889 

375 

19-3649 

7-211 

423 

20-5669 

7-506 

328 

18-1107 

6-896 

376 

19-3907 

7-217 

424 

20-5912 

7-512 

329 

18-1383 

6-903 

377 

19-4164 

7-224 

425 

20-6155 

7-518 

330 

18-1659 

6-910 

378 

19-4422 

7-230 

426 

20-6397 

7-524 

331 

18-1934 

6-917 

379 

19-4679 

7-236 

427 

20-6639 

7-530 

332 

18-2208 

6-924 

380 

19-4935 

7-243 

428 

20-6881 

7-536 

333 

18-2482 

6-931 

381 

19-5192 

7-249 

429 

20-7123 

7-541 

334 

18-2756  6-938 

382 

19-5448 

7-255 

;  430 

20-7364 

7-547 

335 

18-3030  6-945 

383 

19-5703 

7-262 

431 

20-7605 

7-553 

836  18-3303  6*853 

384 

19-5969 

7-268  1)  438 

20-7848 

7-559 

89b 


SQUARE    AND    CUBE    ROOTS. 


Mo. 

Squire  root. 

Cuberdbt. 

No. 

Squire  root. 

Cube  root. 

No. 

Square  root. 

Cube  root. 

433 

20-8086 

7-565 

481 

21-9317 

7-835 

529 

23- 

8-087 

434 

20-8326 

7-571 

482 

21-9544 

7-840 

530 

23-0217 

8-092 

435 

20-8566 

7-576 

483 

21-9772 

7-846 

531 

23-0434 

8-097 

436 

20-8806 

7-582 

484 

22- 

7-851 

532 

23-0651 

8-102 

437 

20-9045 

7-588 

485 

22-0227 

7-856 

533 

23-0867 

8-107 

438 

20-9284 

7-594 

486 

22-0454 

7-862 

534 

23-1084 

8-112 

439 

20-9^23 

7-600 

487 

22-0680 

7-867 

535 

23-1300 

8-118 

440 

20-9761 

7-605 

488 

22-0907 

7-872 

536 

23-1516 

8-123 

441 

21- 

7-611 

489 

22-1133 

7-878 

537 

2:M732 

8-128 

442 

21-0237 

7-617 

490 

22-1359 

7-883 

538 

23-1948 

8-133 

443 

21-0475 

7-623 

491 

22-1585 

7-889 

539 

23-2103 

8-138 

444 

21-0713 

7-628 

492 

22-1810 

7-894 

540 

23-2379 

8-143 

445 

21-0950 

7-634 

493 

22-2036 

7-899 

541 

23-2594 

8-148 

446 

21-1187 

7-640 

494, 

22-2261 

7-905 

542 

23-2808 

8-153 

447 

21-1423 

7-646 

495 

22-2485 

7-910 

543 

23-3023 

8-158 

448 

21-1660 

7-651 

496 

22-2710 

7-915 

544 

23-3238 

8-163 

449 

21-1896 

7-657 

497 

22-2934 

7-921 

545 

23-3452 

8-168 

450 

21-2132 

7-663 

498 

22-3159 

7-926 

546 

23-3666 

8-173 

451 

21-2367 

7668 

499 

22-3383 

7-931 

547 

23-3880 

8-178 

452 

21-2602 

7-674 

500 

22-3606 

7-937 

548 

23-4093 

8-183 

453 

21-2837 

7-680 

501 

22-3830 

7-942 

549 

23-4307 

8-188 

454 

21-3072 

7-685 

502 

22-4053 

7-947 

550 

23-4520 

8-193 

455 

21-3307 

7-691 

503 

22-4276 

7-952 

551 

23-4733 

8-198 

456 

21-3541 

7-697 

504 

22-4499 

7-958 

552 

23-4946 

8-203 

457 

21-3775 

7-702 

505 

22-4722 

7-963 

5f)3 

23-5159 

8-208 

458 

21-4009 

7-708 

506 

22-4944 

7-968 

554 

23-5372 

8-213 

459 

21-4242 

7-713 

507 

22-5166 

7-973 

555 

23-5584 

8-217 

460 

21-4476 

7-719 

508 

22-5388 

7-979 

556 

23-5796 

8-222 

461 

21-4709 

7-725 

509 

22-5610 

7-984 

557 

23-6008 

8-227 

462 

21-4941 

7-730 

510 

22-5831 

7-989 

558 

23-6220 

8-232 

463 

21-5174 

7-736 

511 

22-6053 

7-994 

559 

23-6431 

8-237 

464 

21-5406 

7-741 

512 

22-6274 

8- 

560 

23-6643 

8-242 

465 

21-5638 

7-747 

513 

22-6405 

8-005 

561 

23-6854 

9-247 

466 

21-5870 

7-752 

514 

22-6715 

8-010 

562 

23-7065 

8-252 

467 

21-6101 

7-758 

515 

22-6936 

8-015 

563 

23-7276 

8-257 

468 

21-6333 

7-763 

516 

22-7156 

8-020 

564 

23-7486 

8-262 

469 

21-6564 

7-769 

517 

22-7376 

8-025 

565 

23-7697 

8-267 

470 

21-6794 

7-774 

518 

22-7596 

8-031 

566 

23-7907 

8-271 

471 

21-7025 

7-780 

519 

22-7815 

8-036 

567 

23-8117 

8-276 

472 

21-7255 

7-785* 

520 

22-8035 

8-041 

568 

23-8327 

8-281 

473 

21-7485 

7-791 

521 

22-8254 

8-046 

569 

23-8537 

8-286 

474 

21-7715 

7-796 

522 

22-8473 

8-051 

570 

23-8746 

8-291 

475 

21-7944 

7-802 

523 

22-8691 

8-056 

571 

23-8956 

8-296 

476 

21-8174 

7-807 

524 

22-8910 

8-062 

572 

23-9165 

8-301 

477 

21-8403 

7-813 

525 

22-9128 

8-067 

573 

23-9374 

8-305 

478 

21-8632 

7-818 

526 

22-9346 

8-072 

574 

23-9582 

8-310 

)479 

21-8860 

7-824 

527 

22-9564 

8-077 

575 

23-9791 

8-315 

480 

21-9089 

7-829 

5^8 

22-9782 

8-082 

576 

24- 

8-320 

SQUARE    AND    CUBE    ROOTS. 


299 


No. 

Square  Tool. 

i 

Cube  root. 

fc 

Square  root. 

Cube  root. 

No. 

S  p»re  root. 

Cub.  mot. 

577 

24  0208 

8-325 

625 

25- 

8-549 

673 

25-9422 

8-763 

578 

24-0416 

8-329 

G2C,  ^5-0199 

8-554 

674 

25-9615 

8-767 

579 

24-0624 

8-334 

<;-,'7, 

26-0399 

07  5 

25-9807 

8-772 

580 

24-0831 

8-339 

628 

25-^)599  8-563 

676 

26- 

8-776 

581 

24-1039 

8-344 

629 

25-0798 

8-5C8  <;-,? 

26-0192 

8-780 

582 

24-1246 

8-349 

630 

25-0998 

8-6T2 

678 

26-0384 

8-785 

583 

24-1453 

8-353 

631 

25-1197 

8-577 

2()-(i:>"/(i 

8-789 

584 

24-1660 

8-358 

632 

25-13!*6 

8-581 

680 

26-0768 

8-793 

585 

24-1867 

8-363 

633 

25-1594 

8-586 

t;si 

26-0959 

8-797 

586 

24-2074 

8-368 

634 

25-1793 

8-590 

26-1151 

8-802 

587 

24-2280 

8-372 

635 

25-1992 

8-595 

683 

26-1342 

8-806 

588 

24-2487 

8-377 

636 

25-2190 

8-599  J684 

26-1533 

8-810 

589 

24-2693 

8-382 

637 

25-2388 

8-604 

685 

26-1725 

8-815 

590 

24-2899 

8-387 

638 

25-2586 

8-608 

IkSli 

26-1916 

8-819 

591 

24-3104 

8-391 

639 

25-2784 

8-613 

687 

26-2106 

8-823 

592 

24-3310 

8-396 

640 

25-2982 

8-617 

688 

26-2297 

8-828 

593 

24-3515 

8-401 

641 

25-3179 

8-622 

689 

26-2488 

8-832 

594 

24-3721 

8-406 

642 

25-3377 

8-626 

690 

26-2678 

8-836 

595 

24-3926 

8-410 

643 

25-3574 

8-631 

61)1 

26-2868 

8-840 

596 

24-4131 

8-415 

644 

25-3771 

8-635  i)!>2 

26-3058 

8-845 

597 

24-1335 

8-420 

645 

25-39B8 

8-640  r,!»:: 

26-3248 

8-849 

698 

24-4540 

8-424 

646 

25-4165 

8-644  694 

26-3438 

8-8M 

599 

24-4744 

8-429 

647 

25-4361 

8-649  695 

26-3628 

8-857 

600 

24-4948 

8-434 

648 

25-4558 

8-653  69<> 

26-3818 

8-862 

601 

24-5153 

8-439 

649 

25-4754 

8-657 

897 

26-4007 

8-866 

602 

24-5356 

8-443 

650 

25-4950 

8-662 

098 

26-4196 

8-870 

603 

24-5560 

8-448 

651 

25-5147 

8-666 

6<J9 

26-4386 

8-874 

604 

24-5764 

8-453 

652 

25-5342 

8-671 

700 

26-4575 

8-879 

605 

24-5967 

8-457 

653 

25-5538 

8-675 

701 

26-4764 

8-883 

606 

24-6170 

8-462 

654 

25-5734 

8-680  702 

26-4952 

8-887 

607 

24-6373 

8-466 

956 

25-5929 

B-684 

703 

26-5141 

8-891 

608 

24-6576 

8-471 

656 

25-6124 

8-688 

704 

26-5329 

8-895 

609 

24-6779 

8-476 

657 

25-6320 

8-693 

705 

26-5518 

8-900 

610 

24-6981 

8-480 

658 

25-6515 

8-697 

706 

26-5706 

8-904 

611 

24-7184 

8-485 

659 

25-6709 

8-702 

707 

26-5894 

8-908 

612 

24-7386 

8-490 

660 

25-6904 

8-706 

708 

26-6082 

8-912 

613 

24-7588 

8-494 

661 

25-7099 

8-710 

709 

26-6270 

8-916 

614 

24-7790 

8-499 

662 

25-7293 

8-715 

710 

26-6458 

8-921 

615 

24-7991 

8-504 

663 

25-7487 

8-719 

711 

26-6645 

8-925 

61G 

24-8193 

8-508 

664 

25-7681 

8-724 

71-2 

26-6833 

8-929 

617 

24-8394 

8-513 

665 

25-7875 

8-728 

713 

26-7020 

8-933 

618 

24-8596 

8-517 

666 

25-8069  8-732  714 

26-7207 

8-937 

619 

24-8797 

8-522 

667 

25-8263  8-737 

715 

26-7394 

8-942 

620 

24-8997  8-527 

668 

25-8456  8-741 

716 

26-7581 

8-945 

621 

24-9198  8-531 

669 

25-8650  8-745 

717 

26-7768 

8-950 

622 

24-9399  8-536 

670 

25-8843  8-750 

718 

26-7955 

8-954 

623 

24-959^ 

671 

25-9036  8-754  719 

26-8  14.1 

8-958 

624 

24-9799  ;  8-545 

672 

25-9229  8-759  720 

20-3328 

8-962  . 

300 


SQUARE    AND    CUBE    ROOTS. 


1 

1  No. 

Square  root. 

Cube  root. 

No. 

Square  root. 

Cube  root. 

No. 

Square  root. 

Cube  root. 

721 

26-8514 

8-966 

769 

27-7308 

9-161 

817 

28-5832 

9-348 

722 

26-8700 

8-971 

770 

27-7488 

9-165 

818 

28-6006 

9-352 

723 

26-8886 

8-975 

771 

27-7668 

9-169 

819 

28-6181 

9-356 

724 

26-9072 

8-979 

772 

27-7848 

9-173 

820 

28-6356 

9-359 

72ft 

26-9258 

8-983 

773 

27-8028 

9-177 

821 

28-6530 

9-363 

726 

26-9443 

8-987 

774 

27-8208 

9-181 

822 

28-6705 

9-367 

727 

26-9629 

8-991 

775 

27-8388 

9-185 

823 

28-6879 

9-371 

728 

26-9814 

8-995 

776 

27-8567 

9-189 

824 

28-7054 

9-375 

729 

27- 

9- 

777 

27-8747 

9-193 

825 

28-7228 

9-378 

730 

27-0185 

9-004 

778 

27-8926 

9-197 

826 

28-7402 

9-382 

731 

27-0370 

9-008 

779 

27-9105 

9-201 

827 

28-7576 

9-386 

732 

27-0554 

9-012 

780 

27-9284 

9-205 

828 

28-7749 

9-390 

733 

27-0739 

9-016 

781 

27-9463 

9-209 

829 

28-7923 

9-394 

734 

27-0924 

9-020 

782 

27-9642 

9-213 

830 

28-8097 

9-397 

73ft 

27-1108 

9-024 

783 

27-9821 

9-216 

831 

28-8270 

9-401 

736 

27-1293 

9-028 

784 

28- 

9-220 

832 

28-8444 

9-405 

737 

27-1477 

9-032 

785 

28-0178 

9-224 

833 

28-8617 

9-409 

738 

27-1661 

9-036 

786 

28-0356 

9-228 

834 

28-8790 

9-412 

739 

27-1845 

9-040 

787 

28-0535 

9-232 

835 

28-8963 

9-416 

740 

27-2029 

9-045 

788 

28-0713 

9-236 

836 

28-9136 

9-420 

741 

27-2213 

9-049 

789 

28-0891 

9-240 

837 

28-9309 

9-424 

742 

27-2396 

9-053 

790 

28-1069 

9-244 

838 

28-9482 

9-427 

743 

27-2580 

9-057 

791 

28-1247 

9-248 

839 

28-9654 

9-431 

744 

27-2763 

9-061 

792 

28-1424 

9-252 

840 

28-9827 

9-435 

745 

27-2946 

9-065 

793 

28-1602 

9-256 

841 

29. 

9-439 

746 

27-3130 

9-069 

794 

28-1780 

9-259 

842 

29-0172 

9-442 

747 

27-3313 

9-073 

795 

28-1957 

9-263 

843 

29-0344 

9-446 

748 

27-3495 

9-077 

796 

28-2134 

9-267 

844 

29-0516 

9-450 

749 

27-3678 

9-081 

797 

28-2311 

9-271 

845 

29-0688 

9-454 

750 

27-3861 

9-085 

798 

28-2488 

9-275 

846 

29-0860 

9-457 

751 

27-4043 

9-089 

799 

28-2665 

9-279 

847 

29-1032 

9-461 

752 

27-4226 

9-093 

800 

28-2842 

9-283 

848 

29-1204 

9-465 

753 

27-4408 

9-097 

801 

28-3019 

9-287 

849 

29-1376 

9-468 

754 

27-4590 

9-101 

802 

28-3196 

9-290 

850 

29-1547 

9-472 

755 

27-4772 

9-105 

803 

28-3372 

9-294 

851 

29-1719 

9-476 

756 

27-4954 

9-109 

804 

28-3548 

9-298 

852 

29-1890 

9-480 

757 

27-5136 

9-113 

805 

28-3725 

9-302 

853 

29-2061 

9-483 

758 

27-5317 

9-117 

806 

28-3901 

9-306 

854 

29-2232 

9-487 

759 

27-5499 

9-121 

807 

28-4077 

9-310 

855 

29-2403 

9-491 

760 

27-5680 

9-125 

808 

28-4253 

9-314 

856 

29-2574 

9-494 

761 

27-5862 

9-129 

809 

28-4429 

9-317 

857 

29-2745 

9-498 

762 

27-6043 

9-133 

810 

28-4604 

9-321 

858 

29-2916 

9-502 

763 

27-6224 

9-137 

811 

28-4780 

9-325 

859 

29-3087 

9-505 

764 

27-6405 

9-141 

812 

28-4956 

9-329 

860 

29-3257 

9-509 

765 

27-6586 

9-145 

813 

28-5131 

9-333 

861 

29-3428 

9-513 

766 

27-6767 

9-149 

814 

28-5306 

9-337 

862 

29-3598 

9-517 

767 

27-6947 

9-153 

815 

28-5482 

9-340 

863 

29-3768 

9-520  j 

768 

27-7128 

9-157 

816 

28-5653 

9-344 

864 

29-3938 

9-524 

SQUARE    AND    CUBE    ROOTS. 


301 


No. 

Square  root. 

"ube  root. 

No. 

Squirt  root. 

Cube  root. 

No. 

S<ji«rp  mot. 

Cube  root. 

865 

29-4108 

9-528 

910 

30-1662 

955 

30-9030 

9-847 

866 

29-4278 

9-531 

911 

30-1827 

9-694 

9S6 

30-9192 

9-851 

867 

29-4448 

9-535 

912 

30-1993 

9-697 

957 

30-9354 

9-854 

868 

29-4618 

9-539 

913 

30-2158 

9-701 

958 

30-9515 

9-857 

869 

29-4788 

9-542 

914 

30-2324 

9-704 

959 

30-9677 

9-861 

870 

29-4957 

9-546 

915 

30-2489 

9-708 

960 

30-9838 

9-864 

871 

29-5127 

9-550 

916 

30-2654 

9-711 

961 

31- 

9-868 

872 

29-5296 

9-553 

917 

30-2820 

9-715 

962 

31-0161 

9-871 

873 

29-5465 

9-557 

918 

30-2985 

9-718 

963 

31-0322 

9-875 

874 

29-5634 

9-561 

919 

30-3150 

9-722 

964 

31-0483 

9-87S 

875 

29-5803 

9-564 

920 

30-3315 

9-725 

965 

31-0644 

9-881 

876 

29-5972 

9-568 

921 

30-3479 

9-729 

966 

31-0805 

9-885 

877 

29-6141 

9-571 

922 

30-3644 

9-732 

967 

31-0966 

9-888 

878 

29-6310 

9-575 

923 

30-3809 

9-736 

968 

31-1126 

9-892 

879 

29-6479 

9-579 

924 

30-3973 

9-739 

969 

31-1287 

9-895 

880 

29-6647 

9-582 

925 

30-4138 

9-743 

970 

31-1448 

9-898 

881 

29-6816 

9-586 

926 

30-4302 

9-746 

971 

31-1608 

9-902 

882 

29-6984 

9-590 

927 

30-4466 

9-750 

972 

31-1769 

9-905 

883 

29-7153 

9-593 

028 

30-4630 

9-753 

973 

31-1929 

9-909 

884 

29-7321 

9-597 

929 

30-4795 

9-757 

974 

31-2089 

9-912 

885 

29-7489 

9-600 

930 

30-4959 

9-761 

975 

31-2249 

9-915 

886 

29-7657 

9-604 

931 

30-5122 

9-764 

976 

31-2409 

9-919 

887 

29-7825 

9-608 

932 

30-5286 

9-767 

977 

31-2569 

9-922 

888 

29-7993 

9-611 

933 

30-5450 

9-771 

978 

31-2729 

9-926 

889 

29-8161 

9-615 

<J34 

30-5614 

9-774 

979 

31-2889 

9-929 

890 

29-8328 

9-619 

935 

30-5777 

9-778 

980 

31-3049 

9-932 

891 

29-8496 

9-622 

936 

30-5941 

9-782 

981 

31-3209 

9-936 

892 

29-8663 

9-626 

937 

30-6104 

9-785 

982 

31-3368 

9-939 

893 

29-8831 

J9-629 

938 

30-6267 

9-788 

983 

31-3528 

9-943 

894 

29-8998 

9-633 

939 

30-6431 

9-792 

984 

31-3687 

9-946 

895 

29-9165 

9-636 

940 

30-6594 

9-795 

985 

31-3847 

9-949 

896 

29-9332 

9-640 

941 

30-6757 

9-799 

986 

31-4006  9-953 

897 

29-9499 

9-644 

942 

30-6920 

9-802 

987 

31-4165  ;  9-956 

898 

29-9666 

9-647 

943 

30-7083 

9-806 

988 

31-4324  9-959 

899 

29-9833 

9-651 

944 

30-7245 

9-809 

989 

31-4483 

9-963 

900 

30- 

9-654 

945 

30-7408 

9-813 

990 

31-4642  9-966 

901 

30-0166 

9-658 

946 

30-7571 

9-816 

991 

31-4801  9-969 

902 

30-0333 

9-662 

947 

30-7733 

9-820 

992 

31-4960  9-973 

903 

30-0499 

9-665 

948 

30-789fi 

9-823 

993 

31-5119  9-976 

904 

30-0665 

9-669 

949 

30-8058 

9-827 

994 

31-5277  9-979 

906 

30-0832 

9-672 

950 

30-8220 

9-830 

995  31-5436  9-983 

906 

30-0998 

9-676 

951 

30-8382 

9-833 

996  31-5594  9-986 

907 

30-1164 

9-679 

952 

30-8544 

9-837 

997 

31-5753  9-989 

908 

30-1330 

9-683 

953 

30-8706 

9-840 

998 

31-5911 

9-993 

|  909  30-1496 

9-686  ||  954 

30-8868 

9-844 

999 

31-6069 

9-996 

302  RECIPES. 

USEFUL  RECIPES  FOR  WORKMEN. 


For  Lead. — Melt  one  part  of  block  tin,  and  when  in  a 
state  of  fusion  add  two  parts  of  lead.  If  a  small  quantity 
of  this,  when  melted,  is  poured  out  upon  the  table,  there 
will,  if  it  be  good,  arise  little  bright  stars  upon  it.  Resin 
should  be  used  with  this  solder. 

For  Tin, — Take  four  parts  of  pewter,  one  of  tin,  and  one 
of  bismuth  ;  melt  them  together,  and  run  them  into  thin 
slips.  Resin  is  also  used  with  this  solder. 

For  Iron. — Good  tough  brass,  with  a  little  borax. 

CEMENTS. 

A  very  strong  glue  is  made  by  adding  some  powdered 
chalk  to  common  glue  when  melted;  and  a  glue  which  will 
resist  the  action  of  water  may  be  formed  by  boiling  one 
pound  of  common  glue  in  two  quarts  (English  measure)  of 
skimmed  milk. 

Turkey  Cement. — Dissolve  five  or  six  bits  of  mastich,  as 
large  as  peas,  in  as  much  spirit  of  wine  as  will  dissolve  it. 
In  another  vessel  dissolve  as  much  isinglass,  (which  has 
been  previously  soaked  in  water  till  it  is  softened  and 
swelled,)  in  one  glass  of  strong  whisky;  add  two  small  bits 
of  gum  galbanum,  or  ammoniacum,  which  must  be  rubbed 
or  ground  till  dissolved,  then  mix  the  whole  by  the  assist- 
ance of  heat.  It  must  be  kept  in  a  stopped  phial,  which 
should  be  set  in  hot  water  when  the  cement  is  to  be  used- 

For  turners,  an  excellent  cement  is  made  by  melting  in 
a  pan  over  the  fire  one  pound  of  resin,  and  when  melted 
add  a  quarter  of  a  pound  of  pitcli :  while  these  are  boiling 
add  brick  dust,  until,  by  dropping  a  little  upon  a  cold  stone, 
you  think  it  hard  enough.  In  winter  it  is  sometimes  found 
necessary  to  add  a  little  tallow. 

In  joining  the  flanches  of  iron  cylinders  or  pipes,  to 
withstand  the  action  of  boiling  water  and  steam,  great  in- 
convenience is  often  felt  by  the  workmen  for  want  of  a 
durable  cement.  The  following  will  be  found  to  answer: 
Boiled  linseed  oil,  litharge,  and  white  lead,  mixed  up  to  a 
proper  consistence,  and  applied  to  each  side  of  a  piece  of 
flannel,  linen,  or  even  pasteboard,  and  then  placed  between 
the  pieces  before  they  are  brought  home,  as  it  is  c  illed,  or 
joined. 


RECIPES.  303 

For  Steam  Engines  an  excellent  cement  is  as  follows  : 
Take  of  sal  ammoniac  two  ounces,  sublimed  sulphur  one 
ounce,  and  cast  iron  filings  or  fine  turnings  one  pound  ;  mix 
them  in  a  mortar,  and  keep  the  powder  dry.  When  it  is 
to  be  used  mix  it  with  twenty  times  its  quantity  of  clean 
iron  turnings,  or  filings,  and  grind  the  whole  in  a  mortar,' 
then  wet  it  with  water,  until  it  becomes  of  a  convenient 
consistence,  when  it  is  to  be  applied  to  the  joint ;  after  a 
time  it  becomes  as  hard  and  strong  as  any  other  part  of  the 
metal. 

LACQUERS    AND    VARNISHES. 

Old  Varnish  is  made  by  pouring,  by  little  and  little,  half 
a  pound  of  drying  oil  on  a  pound  of  melted  copal,  constantly 
stirring  with  a  piece  of  wood.  When  the  copal  is  melted, 
take  the  mixture  off  the  fire  and  add  a  pound  of  Venice 
turpentine ;  then  pass  the  whole  through  a  linen  cloth. 
When  the  varnish  gets  thick  by  keeping,  add  a  little  Venice 
turpentine ;  and  if  it  be  wished  of  a  dark  colour,  amber 
should  be  used  instead  of  copal. 

Black  varnish  for  iron  is  made  of  twelve -parts  of  amber, 
twelve  of  turpentine,  two  of  resin,  two  of  asphaltum,  and 
six  of  drying  oil. 

For  cabinet  work  and  musical  instruments  a  varnish  may 
be  made  thus ;  —  Take  four  ounces  of  gum  sandarack,  two 
ounces  of  lack,  the  same  of  gum  mastich,  and  an  ounce  of 
gum  elemi ;  dissolve  them  in  a  quart  of  the  best  whisky  ; 
the  whole  being  kept  warm  when  they  are  dissolved,  add 
half  a  gill  of  turpentine. 

Lacquer  is  a  varnish  to  be  laid  on  metal,  for  the  purpose 
of  improving  its  appearance  or  preserving  its  polish.  The 
lacquer  is  laid  on  the  surface  of  the  metal  with  a  brush :  the 
metal  must  be  warm,  otherwise  the  lacquer  will  not  spread. 

For  brass  a  good  lacquer  may  be  made  thus : — Take  one 
ounce  of  turmeric  root  ground,  and  half  a  drachm  of  the  best 
dragon's  blood ;  put  them  in  a  pint  of  spirits  of  wine, 
(English  measure,)  and  place  them  in  a  moderate  heat, 
shaking  them  for  several  days.  It  must  then  be  strained 
through  a  linen  cloth,  and  being  put  back  into  the  bottle, 
three  ounces  of  good  seed-lack,  powdered,  must  be  added. 
The  mixture  must  again  be  subjected  to  a  moderate  heat, 
and  shaken  frequently  for  several  days,  when  it  is  again 
strained,  and  corked  tightly  in  a  bottle  for  use. 


304  RECIPES. 

STAINING    WOOD    AND    IVORY. 

Fellow.  Diluted  nitric  acid  will  often  produce  a  fine 
yellow  on  wood  ;  but  sometimes  it  produces  a  brown,  and 
if  used  strong  it  will  seem  nearly  black. 

Red.  A  good  red  may  be  made  by  an  infusion  of  Brazil 
wood  in  stale  urine,  in  the  proportion  of  a  pound  to  a  gal- 
lon This  stain  is  to  be  laid  on  the  wood  boiling  hot ;  and 
before  it  dries  it  should  be  laid  over  with  alum  water.  For 
the  same  purpose  a  solution  of  dragon's  blood  in  spirits  of 
wine  may  also  be  used. 

Mahogany  colour  may  be  produced  by  a  mixture  of  mad- 
der, Brazil  wood,  and  logwood,  dissolved  in  water  and  put 
on  hot.  The  proportions  must  be  varied  by  the  artist  ac- 
cording to  the  tint  required. 

Slack.  Brush  the  wood  several  times  over  with  a  hot 
decoction  of  logwood,  and  then  with  iron  lacquer,  or,  if  this 
cannot  be  had,  a  strong  solution  of  nut  galls. 

Ivory  may  be  stained  blue  thus : — Soak  the  ivory  in  a 
solution  of  verdigris  in  nitric  acid,  which  will  make  it 
green,  then  dip  it  into  a  solution  of  pearlash  boiling  hot, 
and  it  will  turn  blue. 

To  stain  ivory  black  the  same  process  as  for  wood  may 
be  employed. 

Purple  may  be  produced  by  soaking  the  ivory  in  a  solu- 
tion of  sal  ammoniac  into  four  times  its  weight  of  nitrous 
acid. 

To  make  Edge-Tools  from  Cast  Steel  and  Iron. — This 
method  consists  in  fixing  a  clean  piece  of  wrought  iron, 
brought  to  a  welding  heat,  in  the  centre  of  a  mould,  and 
then  pouring  in  melted  steel,  so  as  entirely  to  envelope  the 
iron ;  and  then  forging  the  mass  into  the  shape  required. 

To  colour  Steel  Slue. — The  Steel  must  be  finely  polish- 
ed on  its  surface,  and  then  exposed  to  a  uniform  degree 
of  heat.  Accordingly,  there  are  three  ways  of  colouring : 
first,  by  a  flame  producing  no  soot,  as  spirit  of  wine ;  se- 
condly, by  a  hot  plate  of  iron  ;  and  thirdly,  by  wood  ashes. 
As  a  very  regular  degree  of  heat  is  necessary,  wood  ashes 
for  fine  work  bears  the  preference.  The  work  must  be 
covered  over  with  them,  and  carefully  watched ;  when  the 
colour  is  sufficiently  heightened,  the  work  is  perfect.  This 
colour  is  occasionally  taken  ofF  with  a  very  dilute  marina 
acid 


RECIPES  305 

To  distinguish  Steel  from  Iron. — The  principal  cha- 
as  Aers  by  which  steel  may  be  distinguished  from  iron  are 
GS  follow : — 

1.  After  being  polished,  steel  appears  of  a  whiter,  light 
JSty  hue,  without  the  blue  cast  exhibited  by  iron.     It  also 
takes  a  higher  polish. 

2.  The  hardest  steel,  when  not  annealed,  appears  granu- 
lated, but  dull,  and  without  shining  fibres. 

3.  When  steeped  in  acids,  the  harder  the  steel  is  of  a 
darker  hue  is  its  surface. 

4.  Steel  is  not  so  much  inclined  to  rust  as  iron. 

5.  In  general,  steel  has  a  greater  specific  gravity. 

6.  By  being  hardened  and  wrought,  it  may  be  rendered 
much  more  elastic  than  iron. 

7.  It  is  not  attracted  so  strongly  by  tke  magnet  as  soft 
iron.   It  likewise  acquires  magnetic  properties  more  slowly, 
but  retains  them  longer,  for  which  reason  steel  is  used  in 
making  needles  for  compasses,  and  artificial  magnets. 

8.  Steel  is  ignited  sooner,  and  fuses  with  less  degree  of 
heat  than  malleable  iron,  which  can  scarcely  be  made  to 
fuse  without  the  addition  of  powdered  charcoal ;  Jby  which 
it  is  converted  into  steel,  and  afterwards  into  crude  iron. 

9.  Polished  steel  is  sooner  tinged  by  heat,  and  that  with 
higher  colours,  than  iron. 

10.  In  a  calcining  heat,  it  suffers  less  loss  by  burning 
than  soft  iron  does  in  the  same  heat  and  the  same  time.    In 
calcination  a  light  blue  flame  hovers  over  the  steel,  either 
with  or  without  a  sulphureous  odour. 

11.  The  scales  of  steel  are  harder  and  sharper  than  those 
of  iron  ;  and  consequently  more  fit  for  polishing  with. 

12.  In  a  white  heat,  when  exposed  to  the  blast  of  the 
bellows  among  the  coals,  it  begins  to  sweat,  wet,  or  melt, 
partly  with  light-coloured  and  bright,  and  partly  with  red 
sparkles,  but  less  crackling  than  those  of  iron.     In  a  melt- 
ing heat,  too,  it  consumes  faster. 

13.  In  the  vitriolic  nitrous,  and  other  acids,  steel  is  vio- 
lently attacked,  but  is  longer  in  dissolving  than  iron.  After 
maceration,  according  as  it  is  softer  or  harder,  it  appears  of 
8  lighter  grey  or  darker  colour ;  while  iron,  on  the  other 
band,  is  white. 

26* 


UCS8  LIBRARY 


A     000  606  496     8 


